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NEW  AND  COMPLETE  SYSTEM 


OF 


ARITHMETICS 


COMPOSED    FOR    THE 


USE  OF  THE  CITIZENS  OF  THE  UNITED  STATES. 


BY  NICOLAS  PIKE,  A.M.  A.  A.  S. 


/  for  tie   Use  of  Sd-oelt,  (tinder    the   Direction  of  tbs  Author)  from 
the  Third   Octivo    Edition,  tvbich    was   revised.,  corrtflcJ,  and  im- 
mid  more  particularly  adapted  to  the  Federal  Currency ^ 

BY  NATHANIiiL   LORD,  A.M. 


PUBLISHED  BY  THOMAS   ^  ANDREWS, 

SOLD    AT    THEIR    BOOKSTORE,    NO.    -15.    N' ^XS'^URY-STR  SEJ. 


. .  rr/A'7'AiTi- STREET. 


ICT  OF  MASSACHUSETTS—  TO  WIT. 


BE  IT  REMEMBERED,  that  on  the  seventh  day 
of  November,  in  the  the  thirty  fourth  Year  of  the  Independence  of 
the  United  States  of  America,  THOMAS  &  ANDREWS  of  the  faic  Dif- 
tria,  have  depofited  in  this  Office  the  Title  of  a  Book,  the  Right 
whereof  they  claim  as  proprietors  in  the  Words  following,  to  ivit  :  — 

«  A  new  and  complete  fyftesn  of  Arithmetick,  compofed  for  the  ufe 
of  the  citizens  of  the  United  States.  By  Nicolas  Pike,  A.  M.  A.  A.  S. 
,'  jvc  ;th  edition,  abridged  for  the  ufe  of  fchools,  (under  the  direction 
Author)  from  the  third  odtavo  edition,  which  was  revifed,  cor- 
rected and  improved,  and  more  particularly  adapted  to  the  federal 
currency,  By  Nathaniel  Lord,  A.  M." 

In  Conformity  to  the  Act  of  the  Congrefs  Of  the  United  States* 
intitled,  "  An  A&  for  the  Encouragement  of  Learning,  by  fecuring 
the  Copies  of  Maps,  Charts  and  Books,  to  the  Authors  and  Proprie- 
tors of  fuch  Copies,  during  the  Times  therein  mentioned;"  and  alfo 
to  an  AcSl  intitled,  "  An  Ac5l  fupplementary  to  an  A&,  intitled,  An 
Ad;  for  the  Encouragement  of  Learning,  by  fecuring  the  Copies  of 
Maps,  Charts  and  Books,  to  the  Authors  and  Proprietors  of  fuch 
Copies  during  the  times  therein  mentioned  ;  and  extending  the  Ben- 
e-'its  thereof  to  the  Arts  of  Designing,  Engraving  and"  Etching  Hif- 
al  and  other  Prints."  WM.  S.  SHAW, 

Ckrk  of.  tke  Difiritt  of  Ma/achufdts. 


PREFACE 

TO  THE  THIRD  OCTAVO  EDITION,  FROM  WHICH  THIS 
ABRIDGMENT    IS  MADE. 


THE  demand  for  this  work  dill  continuing,  not- 
{landing  the  publication  of  other  works  on  Arithriielfck 
and  the  higher  branches  oF  the  Mathemeiicks,  is  evidence 
of  its  intrinfick  merit,  and  has  induced  the  Proprietors  of 
the  copy-right  to  prefent  the  publick  with  a  new  ana  im- 
proved Edition. 

Application  was  made  to  the  Author,  requeuing  him  to 
revife  and  improve   the  xvork  for  a  new  Edition  ;  but  he 
declined  on  account  of  want  of  health,  and  the    G 
man,  whom  we  employed,  was  engaged  by  the    Au 
confent.and  improved  and  corrected  the  work  agreeably  to 
his  direclion  and  advice. 

The  moft  important  improvement  in  this  Edition,  is  the 
introduction  of  examples  in  the  Federal  Currency  under 
each  rule  ;  and  while  this  was  conttdered  neceffary,  in  ^i  • 
der  to  extend  the  knowledge  and  life  of  that  currency,  it; 
was  thought  important  not  tojgpit  examples  in  pounds, 
fhillings  and  pence,  which  are,  and  will  continue  tc 
the  bafis  of  many  arithmetical  queltions  ;  and  therefore 
an  acquaintance  with  them  will  always  be  ufcfu.1. 

Mr.  NATHANIEL  LORD.  3^,  of  IptVich,  the  Gentle. 
man  employed  to  correct  and  improve  the  work,  has  be- 
(lowed  much  attention  upon  ir,  and  has  received  from 
Mr.  PIKE  all  the  information  and  advice  he  delired.  The 
manner  in  which  Mr.  LORD  has  executed  the  taik  entrufted 
to  him,  will,  we  hope,  gain  additional  reputation  1 
work,  and  entitle  him  to  the  thanks  of  the  publick. 

•THOZvf  ASiC:?  ANDREWS. 

Boflon9  dpril,  1808. 


OF  MASSACHUSETTS— TO  WIT. 

BE  IT  REMEMBERED,  that  on  the  seventh  day 
of  November,  in  the  the  thirty  fourth  Year  of  the  Independence  of 
the  United  States  of  America,  THOMAS  &  ANDREWS  of  the  faic  Dif- 
triv1!:,  have  depofited  in  this  Office  the  Title  of  a  Book,  the  Right 
whereof  they  claim  as  proprietors  in  the  Words  following,  to  -wit .- — 

"  A  new  and  complete  fyfteni  of  Anthmetick,  compofed  for  the  ufe 
of  the  citizens  of  the  United  States.  By  Nicolas  Pike,  A.  M.  A.  A.  S. 
,'--v.-  ;th  edition,  abridged  for  the  ufe  of  fchools,  (under  the  direction 
author)  from  the  third  odtavo  edition,  which  was  revifed,  cor- 
rected and  improved,  and  more  particularly  adapted  to  the  federal 
currency,  By  Nathaniel  Lord,  A.  M." 

}n  Conformity  to  the  A6t  of  the  Congrefs  of  the  United  States* 
intided,  "  An  Aft  for  the  Encouragement  of  Learning,  by  fecuring 
the  Copies  of  Maps,  Charts  and  Books,  to  the  Authors  and  Proprie- 
tors of  fuch  Copies,  during  the  Times  therein  mentioned;"  and  alfo 
to  an  A6t  intitled,  "  An  A6t  fupplementary  to  an  Act,  intitled,  An 
Aet  for  the  Encouragement  of  Learning,  by  fecuring  the  Copies  of 
M.i ps,  Charts  and  Books,  to  the  Authors  and  Proprietors  of  fuch 
Copies  during  the  times  therein  mentioned  ;  and  extending  the  Ben- 
efits thereof  to  the  Arts  of  Designing,  Engraving  and  Etching  Hif- 
l  and  e:her  Prints."  WM.'  S.  SHAW, 

of  tke  Dijlrifl  of  MaffaclufMs. 


PREFACE 

TO  THE  THIRD  OCTAVO  EDITION,  FROM  WHICH  THIS 
ABRIDGMENT    IS  MADE. 


THE  demand  for  this  work  dill  continuing,  not-- 
Handing the  publication  of  other  works  on  Arithitictfck 
and  the  higher  branches  of  the  Matheroeticks,  is  evidence 
of  its  intrinfick  merit,  and  has  induced  the  Proprietors  of 
the  copy-right  to  prefent  the  publick  with  a  new  and  im- 
proved Edition. 

Application  was  made  to  the  Author,  requefting  him  to 
revife  and  improve   the  work  for  a  new  Edition  j  but  lie 
declined  on  account  of  want  of  health,  and  the    G 
man,  whom  we  employed,  was  engaged  by  the    Au 
confent.and  improved  and  corrected  the  work  agreeably  to 
his  direction  and  advice. 

The  moft  important  improvement  in  this  Edition,  h  the 
introduction  of  examples  in  the  Federal  Currency  under 
each  rine  ;  and  while  this  was  con(idered  neceffary,  in  ci- 
der to  extend  the  knowledge  and  life  of  that  currency,  it, 
was  thought  important  not  to  j^mit  examples  in  pounds 
fhillings  and  pence,  which  are,  and  will  continue  t 
the  ban's  of  many  arithmetical  queftions  j  and  therefore 
an  acquaintance  xvith  them  will  always  be  uilful. 

Mr. 'NATHANIEL    LORD,   $d,   of    IpiVich,   ihe  Gentle- 
man employed  to  correct  and  improve  the  \vork,    h;: 
(lowed  much  attention  upon  it,  and   has    received  from 
Mr.  PIKE  all  th^  information  and  advice  he  delired.     The 
manner  in  which  Mr.  LOR.D  has  executed  the  talk  entrusted 
to  him,  wi'-l,  we  hope,  gain  additional   reputation    FOJ 
work,  and  entitle  him  to  the  thanks  of  the  publick. 

THOMAS*:?  ANDREWS. 

Bofon,  djprit,  1808. 


EXPLANATION*  OF  THE  CHARACTERS 

UJ;.D    IN    TH  :  1SE. 

!THE  fign  of  eqq  :  iv  :  as,    T 2-  pence  =  i 
(hilling,  fig  nines  th  •    .    -•  to 

one  {Hilling  ;  and,  tat    \vhate\ 

precedes  it  ib  e<  JVTS. 

The  fign  of  A<  as,    5+5  =  10. 

5  added  to  5  is  equal  to  10.      Read  5  plus 
or  5  more  5  equal  to  10. 

f"  The  fign  of  Subtraction :  as,  12 — 4=8, 
\  that  is,  i  2  leffened  by  4  is  equal  to  8, or  4  from 
I  2  and  8  remains.  Read  1 2  minus  4,  or  12 
(jiefs  4  equal  to  8. 

fign  of  Multiplication  :  as,  6X5—3-5 
that  is,  6  multiplied  by  5  is  equal  to  30. 
Read  6  into  5  equal  to  30. 

f     The  fign  of  Dlvifiou  :  as,  30-^-5=6,  t:i:it 
•J  is,  30  divided  by  5  is  equal  to  6.     Read  30 
6)3C(5    tby  5  equal  to  6. 

f      Numbers  placed  fraction-wife  do    a". 

inote  divifion,  the  numerator,   or  upper 
ber  being  the  dividend,  and  the-   dcnorni  : 
875 
25        j  or   lower  number,  the  divifor  •„  thus, is 

I  25 

Lthefameas  875-^-25=35. 

The  fign  of  proportion,    thus,  2    :    4    :: 
:    1 6,  that  is,  as  2  is  to  4  fo  is  8  to  16. 

•H-  .Signifies  Geometrical  ProgrcfT.  >a. 

f  Shews  that  the-  difference  between  -2  and 

(  9  added  to  6  is  equal  to  13.  Re. id  9  rni- 

9 — 2+6=13  T  rms  2  P'U5  ^  equal  to  13  I  th,iE  tV.e 

|  line  atop  (called  a  Vinculum}  conned.,  all 
numbers  over  which  it  is  drawn. 


5      SigiiinV;  th.it  the   futn  of  3  and  9  taken 

12 — 34-5=4   £from  12  Icavis  or  is  cq!i.i!  t •:>  4, 

A    2 


EXPLANATION  OF  CHARACTERS. 

Signifies  the  fecond  power,  or  Square. 
Signifies  the  third  power,  or  Cube. 

f      Prefixed  to  any  number  or  quantity,  figni- 
j  fies  thit  the  fquare  root  of  that   number   is 
L  I  required.     It  likewife   (as  alfb   the  character 
\/  or      ['  <j  for  any  other  root)   (lands   for  the  expreiiioa 
of  the  root  of  that   number   or    quantity  to 
which    it    is     prefixed.       As    \f$6—69    and 

-V/io8+3J=!2,    or  36|x=6,  &c. 

f      Prefixed  to  any  number,  fignifiss   tliat  the 
|  cube  root  of  that  number  is  required  or  ex- 

6=i5,  and 


f  —  or 


ARITHMETICS 


Art  or  Science  of  corrvv.it- 

ing  by  numbers,  and  is  cornpTiftd  under  (rtt  pr •:•'•.- 
cipal  or  fundatijental  Rn'tr.  v:/.  N?ic.ti,:-i.  or  Numeration^ 
Addition,  .,-/,  'Multiplicatioif&Ti&i}iv!/i_98. 


NUMERATION 

Teac-hes  the  different  value  of  figures  by  their  d;lK rent 
places,  and  to  read  or  write  any  fum  or  number  by  rheie 
ten  characters,  o,  I,  2,  3,  4,  5,  6,  7,  8.  9.  c  is  called  a 
cypher,  and  all  the  refl  are  called  H^nres  or  di 

2.  Befides  the  fimple  va'ue  of  figures  is   abavc  noted, 
they  have  each   a  local  value,  according  to  th^  : 

law  ;  viz.  In  a  combination  of  figures,  reckoi 
right  to  kft,  the  figure,  ia  the  firit  pi  ice,  repreients  >:s 
primitive  limple  value  ;  that  in  the  f-cond  place,  ten 
times  its  fixnple  value,  and  ib  en  ;  the  value  of  the  figure, 
in  each  fucceeding  place,  being  ten  times  the  value  of  it, 
in  that  imrceuiiteiy  preecd:ng  it. 

3.  The  values ef  the  places  arc  !  ^.c^or.r-ng  to 
tliei'  order  :  the  firi'l  is  den.                  ihe  place  of  units  ; 
the  fecond,  tens  ;  the  third,  hundreds ;    and  fo  on,  as  in 
the  Table.      Thus,  in  the  runiHer    34^7:     7,   in   the  Hi  ft 
pi  a  >?,  fig'iiifics  only  f-.vcn  ;   6,  in  the  Icc-md    pl.ice,   figni- 
iies  6  lens,  or  fixty  ;  4,  in  the  third  place,  four  hundre'd  ; 
3.    in  the  fourth  place,  three  thoufind  ;  and   the  whole, 
tak»n  cogether,  is  read  thus  ;    ihrss    thoufand  four  hun- 
dred and  {:\;y  (even. 

4    A  cypher,  though  of  no  ngn-fi  ration  itfclf,  yet  pof- 
fefies  a  place,  und,  vhsiiiet  ca  the  righi  h.iad  oi  f: 


8  NUMERATION. 

in  whole  numbers,  increafes  their  value  in  he  fame  ten- 
fold proportion  ;  thus,  9  figniries  only  nine,-,  but,  if  a  cy- 
pher be  placed  on  its  right  hand,  thus,  90,  it  then  be- 
comes ninety. 

To  enumerate  any  parcel  of  figures,  obferve  the   fol- 
io wirfg  RULE. 

Fir  it  Commit  the  words  at  the  head  of  the  Table, 
viz  units,  tens>  hundreds,  3cc.  to  memory  ;  then,  to  the 
fimple  value  of  .each  figure,  join  the  name  of  its  place, 
beginning  at  the  left  hand  and  reading  towards  the  r'ght. 
More  particular  l\  —  K  Place  a  dot  under  the  right  hand 
figure  of  the  2-.1,  4th,  6th  8th,  &c  half  periods,  and  the 
figure  over  i'uch  dot  will,  univeiikUy,  have  the  name  of 
thoufands.  —  2  Place  the  figures  i,  2,  3,  4,  &c.  as  indi- 
ces, over  the  2ci,  30,  4th,  &c  period  :  Thefe  indices  will 
then  (how  the  number  of  times  the  millions  are  involved' 
—  the  figure  under  J  hearing  the  name  of  millions,  that 
under  2,  the  name  of  billions,  (or  millions  of  millions) 
that  under  3,  trillions,  (or  millions  of  millions  of  mill- 
ions.) 

EXAMPLE. 

Sextill.      Quintill.     QuatrilL    Trillions.    Billions.    Millions.     Units. 


a     a     a     a     a     a 

<a  <a  '"•i  c>  c>  c> 


NOTE  t  .  Billions  is  fubflituted  for  millions  of  mill- 
ions ;  Trillions,  for  millions  of  millions  of  millions  ; 
Quatrillions  for  millions  of  millions  of  millions  of  mill- 
ions. 

Quintillions,  Sextillions,  Septillions,  Oflillions,  Nonill- 
ions,  Deciilions,  Up-deciiiions,  Duodecillions,  &c.  anfwer 
to  millions  ib  ot'caa  involved  as  their  indices  refpedtively 
denote. 

NOTE  2.  The  ri  ;ht:  hand  figure  of  each  half  period  has 
i-..ice  of  units,  of  .hat  half  period  ;  the  an'Jdle  one, 
that  of  tens,  and  ths  left  haiui  one,  th.it  of  hundreds. 


NUMERATION. 


9 


Arn  ICATION. 
Write  down,  in  proper  enures,  the  following  numbers. 

Fifteen      -  __._..          -15 

T\V("  hundred  And   (event v  nine     .... 
Three-  thoufand  four  hundred  snd>three 
Thirty  fs.veri  rhoufand  ii"t:  hundred  -ind  fixcy  feven   - 
Four  hundred  one  thonf-uvJ  fud  rwfiiy  eigiit 
Nine  millions  Irvcnty  two  thouf-ind  and  two  huiidrcd 
Fifty  five  million*,  three  hundred  iii.'ie  thjufand  and 
nine 


•  279 
-    3403 
•    37567 
401038 
9071200 

5,5309009. 


Eight  hundred  millions  forty  four  thoufand  and  ? 
fifty  live 

Two  thouftnd  five  liut)  fired  and  forty  three  rmiiuui.O 
four  hundred  and  thirty  one  thouiand  It  vca  hun-  > 
drcd  and  two  J 


800044055 


454343170? 


Write  clown,  in  words  at  length,  the  following  number^ 


4^7       709040         3 

1,7       3010       879096    '  84094007       49163189186 

1,29     76506     4091875     690748591.     5000984227^. 


Notation  by  Ren: an  Letters. 


J.  One. 

II.  Two. 

III.  Three. 

IV.  Four. 

V.  Five. 
VI    Six. 

VII.  Seven. 

VIII.  Eight. 

IX.  Nine. 

X.  Tea. 

XI-  Eleven. 

XII-  Twelve. 
XIII.  Thirteen 
XLV.  Fourteen. 
XV.  Fifteen. 
XVI    Six*en. 
XVII.  Seventeen. 
XVIII   Eighteen. 

XIX.  Nineteen. 

XX.  Twenty. 
.  Thirty. 


XL    Forty. 
1,.   Fifty. 
LX.   Sixty. 
LXX.  Seventy. 
LXXX.   Eighty. 
XC.  Ninety.' 

C.  Hundred. 

CC.  Two  Hundred. 
CCC.  Three  Hundred. 
CCCC.   Four  Hundred. 

D.  or  13.   Five  Hundred, 
DC.   Six  Hundred. 
DCC.   Seven  Hundred. 
DCCC.   Eight  Hundred. 
DCCCC.   Nine  Hundred. 
M.  or  CIo.  One  Thoufand. 
133.    Five  Thoufand. 

1  033    Fifty  Thoufand. 

-   Five  Hnnt!red  Thouf. 


MDCCC1X.  One    Thoufand  Eigh 
Hundred  and  Nine, 


io  SIMPLE  ADDITION. 

A  lefs  literal  number,  placed  after  a  greater,  always 
augments  the  value  of  the  greater  ;  if  put  before,  it  di- 
niiniihes  it.  Thus,  VI  is  6  ;  IV  is  4  ;  XI  is  i  r  ;  IX  is 

9,  Sec. 


ADDITION 

IS  the  putting  together  of  two  or  more  numbers,  or 
fums,  to  make  them  one  total,  or  whole  mm. 


SIMPLE  ADDITION 

\ 

Is  the  adding  of  feveral  integers  or  whole  numbers  to- 
gether, which  are  all  of  one  kind,  or  fort  ;  as,  7  pounds> 
12  pounds.  an*i  20  pounds,  being  added  together,  their 
aggregate,  or  fum  total,  is  39  pounds. 

RULE. 

Having  placed  uiits  under  units,  tens  under  tens,  &c. 
draw  a  line  underneath,  and  begin  with  the  units  :  After 
adding  up  every  figure  in  that  column,  confider  how  ma- 
ny tens  are  contained  in  their  -fum,  and  placing  the  excefs 
under  the  units,  carry  fo  many  as  you  have  tens,  to  the 
next  column  of  tens :  Proceed  in  the  f '.me  manner  through 
every  column  or  row,  and  fet  down  the  whole  amount  of 
thfc  laft  row. 

PROOF, 

Begin  at  the  top  of  the  lum,  and  reckon  the  figures 
downwards,  in  the  fame  manner  as  they  were  added  up- 
wards, and,  if  it  be  right,  this  aggregate  will  be  equal  to 
the  firft.  Or,  cut  off  the  upper  line  of  figures,  and  find 
the  amount  of  the  reii  ;  then  if  the  amount  and  upper 
line,  when  added,  be  equal  to  the  fum  total,  the  work,  fs 
fuppofed  to  be  right. 


SIMPLE  ADDITION. 

ADDITION  and  SUBTRACTION  TABLE. 


II 


:      I 

2  1     3  '    4l     9 

6|     7 

!     8  j     9  i   10 

II 

121 

.     2 

4  1     U     6       7 

8  i     9 

r  o  -i  1  1  _  }  1  2 

[j 

H 

51     *l      ?  1     8 

9  I    •" 

II    |     !  2    J     IS{ 

'4 

•<r 

4 

6  i     7  |         19 

10  |    i: 

2  ,   -3  i  14 

5 

r6 

f  5< 

7  ;    M    ,  I  «o 

M      |        2 

M  :    14  «  15 

1  6 

i? 

6 

*|    9|  ,.      H 

2      1-5 

>f  1  =5  !  '6 

(  - 

18 

:     7 

;!  i-  i  !•  |.i  2 

f^  |   '4 

15  |  16  '  17 

18 

»9 

;   8 

10    |     <  .     |    12    !     '3 

'4  i  »5 

1  6  ;  i  7  i   '  U 

[9 

20 

i  a 

M  ! 

'5  i  IP 

17   1    IS    I    jg 

20 

21 

IQJ 

'3  i   '4  1  '5 

16)   .7 

18  !  19  |  20 

2. 

22 

When  you  would  add  two  number.-,  look  one  of  them 
in  the  left  hand  column,  and  the  other  atop,  and  in  the 
common  angle  of  meeting,  or  at  the  right  hand  of  the 
fir  ft.  and  under  the  fecond,  you  will  find  the  fum — as,  5 
and  8  is  13. 

When  you  would  fubtraft  ;  find  the  number  to  befub- 
tracled  in  the  left  h.ind  column,  run  your  eye  along  to 
the  right  hand  till  you  find  the  number  from  which  it  is 
to  be  taken,  and  right  over  it,  atop,  you  will  find  the  dif- 
ference— as  8  taken  from  13  leaves  5. 
% 

EXAMPLES. 


I. 

2. 

3- 

4- 

5' 

6. 

£• 

Ib. 

Cv/t. 

Miles. 

Yards. 

£. 

I 

12 

1*3 

I234 

12345 

987654321 

2 

34 

456 

5678 

67890 

123456789 

3 

59 

789 

>9« 

98765 

234567891 

4 

7« 

12 

7654 

43210 

345678910 

5 

90 

345 

32  to 

^345 

456789123 

6 

i 

678 

69 

67890 

567*79287 

7 

23 

901 

4713 

74100 

678^00028 

8 

45 

2  q.l 

J3r 

64786 

789.100690 

9 

67 

567 

9128 

19876 

548769138 

iz  SIMPLE  SUBTRACTION 


f- 

y 

8. 

9- 

1234567 

1234567 

67 

2345678 

723456 

123 

3456789 

245  65 

45-66 

4567890 

4566 

89064 

5678109 

333 

654321 

'6789098 

90 

1234567 

10. 

1234567 
9876*43 

2  «O2b"6? 

432 '234 
5682098 

654.321  8 


SUBTRACTION 

TEACHES  to  tb-ke  a  left  number  from  a  greater,  to 
•"find  a  third,  (hewing  the  inequality,  excefs  or  difference 
between  the  given  numbers  ;  and  it  is  both  fimple  and 
compound. 

SIMPLE  SUBTRACTION 

Teaches  to  find  the  difference  between  any  two  num- 
bers, which  are  of  a  like  kind. 

RULE. 

Place  the  larger  number  uppermofl:,  and  the  lefs  under- 
neath, fo  that   units  may  ft  and  under  units,   ttns  ur-der 
tens,  &c.  then  ('rawing  a  line  underneath,,  begin  with  the 
nni'.s,  -j-  ii  fubir-:cl  the  lower  from  the  upper  figure,  and 
fet  down  the  r,  ni:Jrtder  ;  hue  if  the  lower  figure  be  i?reat- 
,n  the  upper,  l.-.)uov.-  *zn  and  fubtracl  the  lover  fig- 
urs  therefro >.>  :   To   this  di'ff-.rence  add  the  upper  figure, 
hcnig  fct   do  vi,,    y.  M    mtiCt  .tdd   one  to   the   ten's 
pl.-vCi  of  it.  iv»r   tb:-.i    which  you  borrowed  ; 

and  i-  ih  tnc  whole. 


iflion,  add  the  re- 
iim,  if    -he 

,Vie  .,,_. „-.  .        .      ,   ..   ..Qf> 

C-  ,,,'.-  t,       ,.:f 

••  '. .  ;    .  •  I).-- 

' 


SIMPLE  MULTIPLICATION.  13 

EXAMPLES. 

i.         2.           3.             4-               3-  *>. 

£        £         Miles.     Yards.         Feet.  Cwt. 

From  25       305       4670       58934  879647  9187641 

Take  12       103       4020         6182  164348  9^43 

Rem. 
Proof. 


7.  8.  9. 

100200300400500600700800900       10000          1000000 
11023045067089         9999  i 


MULTIPLICATION 

MAY  be  accounted  the  mod  ferviceable  rule  in  Arith- 
metick.  It  teaches  how  to  increafe  the  greater  of  two 
numbers  given,  as  often  as  there  are  units  in  the  lefs  ; 
performs  the  work  of  many  additions  in  the  moll  compen- 
dious manner  :  brings  numbers  of  great  denominations 
into  fmall,  as  pounds  into  (hillings,  pence,  or  farthings, 
&c  ;  and,  by  knowing  the  value  of  one  thing,  we  find 
the  value  of  rna-ny. 

It  cor,firH  of  three  parts. 

1.  The  Multiplicand,  or  number  given  to  be  multiplied, 
and  commonly  the  large'!  nnmbtr 

2.  The  Multiplier,  or  number  to  multiply  by,  common- 
ly the  lead  number. 

3.  The  Product  is  the  refult  of  the  work,  or  the  anfwer 
to  the  queftion. 

SIMPLE  MULTIPLICATION 

Is  the  multiplying  of  any  two  numbers  together,  with- 
ou'  having  regard  to  their  iignificatioH  ;  as,  7  times  8  is 
0,&c. 

B 


SIMPLE  MULTIPLICATION. 
MULTIPLICATION  and  DIVISION  TABLE. 


I 

2 

3 

4 

5 

6 

7 

8  |   9 

10 

II 

:  ra! 

2 

4 

6|  8 

10 

12 

14 

16  j  18 

20 

22 

24 

3 

6 

9 

12 

15   18 

21 

24  |  27 

3C 

33 

36 

4 

8 

12 

1  6 

20  |  24 

z8 

32  |  36 

40 

44 

48 

5 

ro 

15 

20 

*5  I  30 

35 

4O  |  4< 

50 

55 
66 

60 

6 

1  2 

.8 

24 

30 

36 

42 

48  |  -54 

60 

7 

14 

21 

28 

35 

42 

49 

56  i   63 

70 

77 

84 

-8 

J6 

24  -  32 

40  !  4u 

56 

64  -j  11 

80 

88 

96 

9 

18 

27  |  <6 

45 

54 

63 

72  |  81 

9° 

99 

108 

10 

10 

30 

40 

50 

60 

70 

80  j  90 

IOO 

I  1C 

1  20 

1  1 

22 

33 

44 

55 

66 

77  i  88  |  91 

•  IO  1  121 

134 

,17   ^4 

36 

48 

60 

72 

84  |  96  |  108 

X20  j  132 

144 

To  learn  this  Table  for  Multiplication  ; — Find  your  mul- 
tiplier in  the  left  hand  column,  and  your  multiplicand  a- 
top,  and  in  the  common  angle  of  meeting,  or  againft  your 
multiplier  along  at  the  right  hand*  and  under  your  mul- 
tiplicand, you  will  find  the  product  or  anfwer, 

To  learn  it  for  Divijion  ; — Find  the  divifor  in  the  left 
hand  column,  and  run  your  eye  along  the  row  to  the  right 
hand  until  you  find  the  dividend  :  then  directly  over  the 
dividend,  atop,  you  will  find  the  quotient,  fiiowing  how  of* 
ten  the  divifor  is  contained  in  the  dividend. 

CASE  I. 

When  tbs  multiplier  is  not  more  than  12 — Always  placing 
the  greateft  number  uppermoft,  fet  the  multiplier  under- 
neath, units  under  units,  &c.  and  begin  as  the  Table  dU 
reels,  fetting  down  the  unit  figure  under  units,  and  carry- 
ing the  tens  to  the  next  place,  in  all  refpects  as  in  fimple 
addition. 

PROOF. 

Multiply  the  multiplier  by  the  multiplicand. 

EXAMPLES. 

I.  2.  3.  4. 

37934     7693°8     498co/6     76389* 
2345 


Prod. 


SIMPLE  MULTIPLICATION.  r$ 

5.        6.         7.        8. 

5037  4     39'  295g     91  47  5 


9. 
4879567 


ID. 

5864794 


n 


8583478646 


IO 


II 


12 


CASE  II. 

When  the  multiplier  is  more  than  I  2.  —  Multiply  "each  fig- 
tire  in  the  multiplicand  by  every  figure  in  the  multiplier, 
beginning  with  the  units,  and  placing  the  firft  figure  of 
each  product  exactly  under  its  multiplier  :  L-aftly,  add 
thefe  feveral  producis  together,  in  the  fame  order,  as  they 
fland,  and  their  fum  will  be  their  total  product. 

EXAMPLES. 


i. 

6357534 
47 

44502738 
25430136 

Prod.  298804098- 


2. 

8324629 
59 


3. 


76 


4. 

647906 
4873 


5. 

760483 
9152 


6. 

9160-584 
6375 


3157245938     6959940416     631589640000 


CA&E   III. 

When  the  multiplier  is  a  compete  number,  that  is,  i}jben  it 
ts  produced  by  the  multiplication  of  any  tw}  numbers  in  the 


*6  SIMPLE  MULTIPLICATION. 

Tails — .Multiply  the  multiplicand  by  one  of  thofe  figures 
firil,and  that  produft  by  the  other  ;  and  the  laft  product 
will  be  the  total  required. 

EXAMPLES. 

I.  2.  $. 

Multiply  59375  by  35         39187  by  48        91632  by  56 
7 

7X5=35 

415625 

5 


2078125 


CASE    IV. 


When  there  are  cyphers  on  the  right  hand  of  either  the  multi- 
plicand, or  multiplier,  or  both — Neglect  thole  cyphers  :  then 
place  the  fignificant  figures  under  one  another,  ancl  mul- 
tiply by  them  only  j  add  them  together,  as  before  direct- 
ed, and  place  to  the  right  hand  as  many  cyphers  as  there 
are  in  both  the  factors. 


3- 

930137000 
9500 


Prod.  380296000   306144000   8836301500000,: 


CASE  V. 

To  multiply  by  10,  TOO,  1000,  &c. — Set  down  the  mul- 
tiplicand underneath,  and  join  the  cyphers  in  your  multi- 
plier to  the  right  hand  of  them.'* 


*  This  is  evident  from  the  nature  of  numbers,  iince  every  cy- 
pher annexed  to  the  right  of  a  number  increafes  the  value  o/  that 
in  a  tenfold  proportion, 


SIMPLE  DIVISION.  17 

EXAMPLES. 

t.      2.      3.        4. 
57935    84935    613975    8473965 

IO  I  CO  10CO  10OCQ 


Prod, 


DIVISION 

TEACHES  to  feparate  any  number,  or  quantity  given, 
into  any  number  of  parts  aligned  ;  or  to  find  how  often 
one  number  is  contained  in  another  ;  or  from  any  two 
numbers  given  to  find  a  third,  which  fhall  confift  of  fo 
many  units,  as  the  one  of  thofe  given  numbers  is  compre- 
hended in  the  other  ;  and  is  a  concife  way  of  perform- 
ing feveral  fubtraclions. 

There  are  four  principal  parts  to  be  noticed  in  Di- 
vifioru  viz; 

j.  The  Dividend,  or  number  given  to  be  divided. 

2.  The  Divifor,  or  number  given  to  divide  by. 

3.  The  Quotient,   or  anfwer  to-  the   queftion,    which 
{Hows  how  often  the  divifor  is  contained  in  the  dividend. 

4.  The  remainder  (which  is    always  lefs  than  the  di- 
vifor, and  of  the  fame  name  with  the  dividend)  is  very 
uncertain,  as  there  is  fometimes  a  remainder,   and  fome 
times  none. 

Divifion  is  both  fimple  and  compound. 

PROOF; 

Multiply  the  divifor  and  quotient  together,  and  add 
the  remainder,  if  there  be  any,  to  the  product  :  If  the 
work  be  right,  that  fum  will  be  equal  to  the  dividend. 


SIMPLE  DiriS ION 

Is  the  dividing  of  one  number  by  another,  without  re- 
gard to  their  values  ;  as,  56,  divided  by  8,  produces  7 
in  the  quotient :  That  is,  8  is  contained  7  times  in  56. 


iS  SIMPLE  DIVISION. 

CASE  I. 

RULE. 

Pirft,  feek  how  many  times  the  divifor  is  contained  in  a 
competent  number  of  the  tirft  figures  of  the  dividend  j 
when  found,  place  the  figure  in  the  quotient  ;  multiply 
the  divifor  by  this  quotient  figure  ;  place  the  produft  un- 
der the  left  hand  figures  of  the  dividend  ;  then  fubtract  it 
j&erefrom,  and  bring  down  the  next  figure  of  the  dividend 
to  the  right  hand  of  the  remainder  :  If,  wh  en  you  have 
brought  down  a  figure  to  the  remainder,  it  is  ftill  lefs  than 
the  divifor,  a  cypher  mud  be  placed  in  the  quotient,  and  a- 
nother  figure  be  brought  down  ;  after  which  you  mull 
feek,  mukiply,  and  fubtracl:,  till  you  have  brought  down 
every  figure  of  the  dividend.* 

EXAMPLES. 
Divifor,  Divid.  Quo. 
3)175817(58605 
'5 

25  Proof. 

24  58605   Quotient. 

—  X3  Divifor  -f  r 

18 


17 


2  Remainder. 

*  When  there  is  no  remainder  to  3  divifion,  the  quotient  is  the 
ablolute  and  perfect  anfwer  to  the  quefrion  ;  but  where  there  is  a 
remainder,  it  may  be  cbferved,  that  it  goe*  fo  much  towards  anoth- 
er time  as  it  approaches  the  divifor  ;  thus,  if  the  remainder  be  half 
the  divifor,  it  will  go  half  of  a  time  more  and  fo  on  ;  in  order,  there- 
fore, to  complete  the  quotient,  pat  the  Life  remainder  to  the  end  of 
it,  above  a  line,  and  ihe  divifor  below  it. 

It  is  fometimes  difficult  to  find  how  often  the  divifor  may  be  had 
in  the  numbers  of  the  ftverai  fleps  of  the  operation  :  The  befl  way 
will  be  to  find  how  often  the  fit  ft  figure  of  the  divifor  may  be  h^d  in 
the  firft,  or  two  iirft  figures  of  the  dividend,  and  the  ani'wer  made 
ieis  by  one  or  two,  i*  generally  the  figure  wanted  ;  i»ist  if,  after  fub- 
trachng  the  produtft  of  the  divifor  and  quotient  i.u.A  the  dividend, 
the  remainder  be  equal  to,  or  ix';eeda  luc  iHvifor,  the  quotient  figure 


SIMPLE  DIVISION,  ,9 

In  this  example,  I  find  that  3,  the  divifor,  cannot  be  con- 
tained in  the  firft  figure  of  the  dividend  ;  therefore,  I  take 
two  figures,  viz.  1 7,  and  inquire  how  often  3  is  contain- 
ed therein,  which  finding  to  be  5  times,  I  place  the  5  in 
the  quotient,  and  multiply  the  divifor  by  it,  fetting  the 
firft  figure  of  the  multiplication  under  the  7  in  the  divi- 
dend, '&c.  I  then  fubtracl:  15  from  17,  and  find  a  remain- 
der of  2,  to  the  right  hand  of  which  I  bring  down  the  next 
figure  of  the  dividend,  viz.  5  ;  then,  I  enquire  how  often 
the  divifor  3  is  contained  in  25,  and,  finding  it  to  be  8 
times,  I  multiply  by  8,  and  proceed  as  before,  till  I  bring 
down  the  i,  when  finding  I  cannot  have  the  divifor  in 
i,  I  place  o  in  the  quotient,  and  bring  down  7  to  the  I,  and 
proceed  as  at  the  firft. 

Obferve,  that  in  multiplying  by  3, 1  add  in  the  2. 


14150 


35)I97'84( 

8. 

3479)483956795( 
i  o.  n. 


2. 


muft  be  increafed  accordingly  ;  or  if  the  produdl  of  the  divifor  and 
quotient  figure  exceed  the  dividend,  then  the  quotient  figure  muft. 
be  proportionally  IcfTeacd, 


!«  SIMPLE  DIVISION. 


i  23456789)121932631  i 

CASE    II. 

When  there  is  one  cypher  or  more  at  the  right  hand  of  the  di- 
vtfor  —  It  or  they  muft  be  cut  off;  alfo  cut  off  the  fame 
number  of  figures  from  the  dividend,  and  then  proceed  as 
in  cafe  firft  :  But  the  figures  which  were  cut  off  from  the 
dividend  muft  be  placed  at  the  right  hand  of  the  remain- 
der. 

EXAMPLES. 

I.  2. 

65  I  00)3794326  |  75(58374      5I93i°oo)8937643|893( 
325 

544  9'7  I  0)47658  |  3( 

520 

243 

195  4. 

-  -      875  |  000)91764789430  j  ooof 
482 

45£ 

276 
260 

1675  Remainder, 
5.  6.  7. 

Quot.    Rem.  Quot.    Rem,.  Quct.  Reas. 

110)958416  1100)76495180  11000)937518391462 

NOTE-  In  dividing  by  10,  100,  1000,  &c.  when  you 
cut  of  as  many  figures  from  the  dividend,  as  there  are 
cyphers  in  the  divifor,  your  work  is  done  ;  thofe  figures 
cut  off  at  the  right  hand,  are  the  remainder,  and  thofe  on 
the  left,  the  quotient,  as  above. 

CASE    III. 
SHORT  DIVISION  //  when  the  divifor  docs  not  exceed  12. 

RULE. 

Firft,  feek  how  often  the  divifor  can  be  had  in  the  firft 
<re;  or  figures  of  the  dividend  j  which,  when  found, 


SIMPLE  DIVISION.  21 

place  in  the  quotient ;  then,  wentdlfy,  multiply  your  divifor 
by  the  figure  placed  in  the  quotient,  and  fubtradt  the  pro- 
duel  from  the  like  number  of  the  left  hand  figures  of  your . 
dividend,  and  the  units  which  remain  muft  be  accounted 
fo  many  tens,  which  you  muft  fuppofe  to  ftand  at  the  left 
hand  of  the  next  figure  in  the  dividend,  and  to  be  reckon- 
ed with  it ;  then,  feek  how  often  ycu  can  have  your  di- 
vifor in  thofe  two  figures  ;  but,  if  nothing  remain,  you 
muft  then  feek  how  often  your  divifor  is  contained  in  the 
next  figure,  or  figures,  and  thus  proceed  till  you  have 
done. 

EXAMPLES. 

Divifor.  Dividend.         2.  3.  4. 

2)7'935          3)519°3         5!63379*         6)8471937 

Quo.    35967—1 


5.  6.  7. 

3)5437846      9)4596*784       11)91843756 


CASE  IV. 

When  the  divifor  is  fuch  a  number,  that  any  two,  01- 
more,  figures  in  the  table,  being  mutiplied  together,  will 
produce  it,  divide  the  given  dividend  by  one  of  thofe  fig- 
ures ;  the  quotient,  thence  arifmg,  by  the  other,  and  fo 
on  :  and  the  laft  quotient  will  be  the  anfwer.* 

*  As  the  learner,  at^>refent,  is  fuppofed  to  be  unacquainted  with 
the  nature  of  Fractions,  and  as  -the  quotient  is  incomplete  without 
die  remainder  —  I  (hall  here  give  a  rule  for  finding  the  remainder 
without  having  recourfe  to  fractions. 

I. 


Multiply  the  quotient  by  the  divifor  ;  fubtraA  the  product  from 
the  dividend,  and  the  refult  \v\.\  he  the  true  remainder 

The  Rule,  which  is  moft  commonly  made  ufc  of,  when  the  divifor 
is  a  com  polite  number,  is, 

RULE  II. 

Multiply  the  laft  remainder  Uy  the  preceding  divifor,  or  laft  but 
one,  and  to  the  product  add  the  preceding  remainder  ;  multiply  this 
ium  by  the  next  preceding  divifor,  and  to  the  product  add  the  next 
preceding  remainder  ;  and  fo  on  till  you  have  gone  through  all  the 
divifors  and  remainders,  to  the  firft, 


22-  TABLES. 

EXAMPLES. 

Firft  Method,  Second  Method.  ' 

9)  1^-6473-  8)196473 

8)2(830  9)24559 

Quo.     2728 — 57  Quo,     27*8 — 57 

In  the/r/?  operation,  in  dividing  bv  9,  3  remains,  and 
by  8,  6  remains  ;  which  being  the  lait  remainder,  I  mul- 
tiply it  by  the  fir  ft  divifor  9.  and  add  in  the  tirft  remain- 
der 3,  and  they  make  57,  the  true  remainder.  In  the 
fccond  method,  dividing  by  8,  I  remains  and  by  9,  7  re- 
mains ;  I  therefore  multiply  7,  the  lail  remainder,  by  8, 
adding  in  the  i,  and  they  make  57,  as  before. 

2.  3.  4. 

*5) '67835        54)93738764          132)3^47369* 


TABLES  IN  COMPOUND    ADDITION. 


i    FEDERAL  MONEY.* 

10  Mills       "J                   TCent      marked  m.  c. 

10  Cents       I                    J  Dime  d, 

10  Dimes      f  make  '    \  Dollar  D. 

fo  Dollars                           ^le  K 


Mills 

10=       i  Cent 
1  00=     10=     i   dime 
1000=  ioc=  ic=  j  dollar. 
1  0000=1000=  100=  1  0=1  eagle. 

*  It  m>ay  be  proper  to  introduce  here  an  account  of  the  Fedci-  Ai 
Money,  as  fettled  by   Congrefs,  the    8th  of  Augtift,   1786,  when  it 


"  That  the  flandard  of  the  United  States  of  America,  for  gold 
and  Clver,  fhall  be  eleven  parts  fine  and  one  part  alloy. 

"  That  the  Money  Unit  of  the  United  States  (being  by  the  Re- 
folveof  Congrefs  of  the  6th  of  July,  1785,  a  Dollar)  fliali  contaih 
of  fine  filver  375  ^  grain?. 


TABLES.  33 

2.    ENGLISH  MONEY. 

4  Farthings   "1  f  Penny     marked     qrs.  d. 

32  Pence  >make  i  <  Shilling  s. 

20  Shillings     J  (.  Pound  '   £  or  L 

Farthings 

4  =       i   Penny 
48  =;     12—1   Shilling 
960  =  240=520:=  i   Pound. 

"  That  the  money  "f  account,  to  correfpond  with  the  divifion  of 
coins,  agreeably  to  the  above  Rtfolve,  proceed  in  a  decimal  ratio, 
agreeably  to  thx  t.orms  and  manner  following,  v;z. 

*•  VI ill.  the  lowed  money  of  account,  of  which  1000  fhall 
be  equal  to  the  federal  dollar,  or  money  unit,  O-OOI 

"  Cent,  the  hightft  copper  .piece,  of  which  100  fhall  be  c. 
qua  1  to  the  federal  dollar,  O-olO 

"  Dime,  the  lowed  Qlver  coin,  of  which   <o*ihail  be  equal 
to  the  dollar,  0*100 

"  Dollar,  the  higheft  filver  coin,  rcoo 

"  That  betwixt  the  dollar  and  the  lowed  copper  coin,  as  filed  by 
a  refolve  of  Congrefr  of  the  6th  of  July,  1785,  there  ftiall  be  three 
filver  coins,  and  one  copper  corn. 

"  I  hat  the  filvtr  coin*  fliall  be  as  follow:  One  coin  containing 
187-Aj^.  grains  of  fine  filver,  tt>be  called  Half Dollar :  One  coin  contain- 
ing 75Tl_y_  grains  of  fine  filver,  to  he  caled  a  Double  Dime  :  And 
cne  coin  containing  37T?^._  grains  of  fine  filver,  to  he  called  a 
Dime. 

"  That  the  two  copper  coins  {hall  be  as  follow  :  One  equal  to  the 
one  hundredth  part  of  the  federal  dollar,  to  be  called  a  Cent  :  and 
one  equal  to  the  two  hundredth  part  of  the  federal  dollar,  to  be  cal- 
led a  Half  Cent. 

"  That  ^l  pounds  Avoirdupois  weight  of  copper  fliall  conftjtutc- 
100  Cents. 

"That  there  fhall  be  two  gold  coins  :  One  containing  Z46T1^^ 
grains  of  fine  gold,  equal  to  10  dollars,  to  be  damped  with  the  im- 
preflion  of  the  American  Eagle,  and  to  be  called  an  Eagle  :  One 
containing  123^^.  grains  of  fine  gold,  equal  to  5  dollars,  to  be 
damped  in  like  manner,  and  to  be  called  a  Half  Eagle. 

"  That  the  mint  price  of  one  pound  Troy  weight  of  uncoined  di- 
ver, eleven  parts  fine  and  one  part  alloy,  fha'l  be  9  dollars,  9  dimesj 
and  a  cents. 

"  That  the  mint  price  of  one  pound  Troy  weight  of  uncoined  gold, 
eleven  parts  fine  and  one  part  alloy,  (hall  be  109  dollars,  7  dimcss 
.and  7  cent*," 


TABLES. 


PENCE  TABLES. 

d.         s.      d. 

d.           s.       d. 

20=1          8 

I2O   =    IO         O 

30  =  2       6 

130  =  10     io 

40  =  3       4 

140  =11       8 

50  =  4       2 

150  =  12       6 

60  =  5       o 

1  60  =   13       4 

70  =  5     10 

IJO   —    14         2 

80  =  6       8 

180  =15       o 

90  =  7       6 

190    =    15        IQ 

100  =8       4 

200  —  16       8 

no  ..=  9       2 

240   =20         0 

s.          d. 

S.                    d. 

I         e=         12 

11       s=       132 

2       =       24 

J2       =       I44 

3     =     36 

13       =       I56 

4    =    48 

14     =.     1  68 

5     =     60 

15     =     180 

6    =     72 

1  6     =     192 

7     =     84 

1  7     =     204 

8     =     96 

18     =     216 

9     =  ic8 

19     =     228 

10      =    120 

20      =       240 

3.  TROY 

WEIGHT.* 

Marked 

24 

20 

Grains                 make 
Pennyweights 

[     Pennyweight,        grs.  pwto 
Ounce                     oz. 

12 

Ounces 

Pound                    jfc  or  lb. 

4.     AVOIRDUPOIS  'WEIGHT.^ 

16 

Drams        make     i  Ounce             marked     dr.     oz. 

16 

Ounces                      i   Pound                                       lb. 

28 

Pounds                     £  of  a  Hundred  weight             qr. 

4 

20 

* 

Quarters                   i   Hund.  weight,  or  i  I2lbs.    Cwt. 
Hundred  weight       i     \>n.                                          T. 

Rv  this  wpicrht    arp    wriorhfH     flnld_  $ilv^>r_   Tpjurli    F.lpf^nari^s 

and  i\\l  Liquors 

NOTE  175  Troy  ounces  are  precifcly  equal  to  191  Avoirdupois 
ounces,  and  175  Troy  pound-  are  equal  to  144  Avoirdupois.  lib 
Xi<  y— 5760  grains,  and  ilh  Avoirdupois  =r  7000  grains. 

•j-  by  Avoirdupois  are  weighed  all  coarfe  aud  droliy  goods  grocery 
aad  chandlery  ware*  ;  bread,  and  all  metals,  except  Gold  and  Silver 


5- 

APOTHECARIES'  WEIGHT.* 

20  Grains 
3  Scruples 
8  Drams 
32  Ounces 

make  i     Scruple 
Dram 
Ounce 
Pound 

marked 

TABLES.  25 


gr-9 
3 


6.    CLOTH  MEASURE. 

2f  Inches              make  i  Nail             marked  in.  na. 

4  Nails,  or  9  Inches  Quarter  of  a  Yard  qr. 

4  Quarters,  or  36  In.  Yard  yd. 
3  Quarters,  or  27  In.  Ell  Flemifh  E.  Fl. 

5  Quarters,  or  45  In.  Ell  Englifh  E.  E. 

6  Quarters,  or  54  In.  Ell  French  E.  Fr. 


7.    LONG  MEASURE.-J- 

3  Barley  corns  make  i  Inch  marked  bar.  in. 
12  Inches  Foot  ft. 

3  Feet  Yard  yd. 

5*  Yards  or  1  61  ft.  Rod,  Perch  or  Pole  pol. 

40  Poles  Furlong  fur. 

8  Furlongs  _Mile  mile. 

Degree  of  a 


69 \  Statute  Miles,  nearly 
360  Degrees 


great  Circle  deg. 

*  A  great  Circle  of 
the  Earth. 


*  -All  the  weights  now  ufcd  by  Apothecaries,  above  grains,  are 
Avoirdupois. 

The  Apothecaries'  pound  and  ounce,  and  the  pound,  and  ounce 
Troy,  are  the  fame,  only  differently  divided  and  fubdivided. 

f  The  ufe  of  Long  Meafure  is  to  meafure  the  difb.nce  of  places 
or  any  other  thing,  where  length  is  confidercd  without  regard  to 
breadth. 

NOTE.  60  geometrical  miles  make  a  degree — 4  inches  a  hand — 5 
feet  a  geometrical  pace,  6  point-  make  i  line,  n  lines  an  inch,  u 
inches  a  foot  and  6  feet  one  French  toife,or  fathom,  equal  to  6  feet  4 
inches,  8-811875  lines  Englifh  meafure.  I  Englifh  foot  equal  tc  II 
inches  3  1154  lines  French.  66  feet,  or  4  poles  make  a  Gunters 
chain,  3  miles  make  a  league, 

c 


2<J  TABLES. 

Ory  in  me  a  fur  ing  Dijlances^ 
7i?o\  Inches  make  i     Link 

55  Links  Pole 

100  Links  Chain 

10  Chains  Furlong 

8  Furlongs  Mile 

8.  TIME.* 

60  Seconds  make  i  Minute  marked  s.ro, 

60  Minutes  Hour  h. 

24  Hours  Day  d. 

7   Days  Week  w. 

4  Weeks  Month  m. 

3  3  Months  i  day  6c  6  hours  Julian  Year.  yr. 

9.     LAND,  or  SQUARE  MEASURE. 

144  Inches  make   i          Square  Foot 

9  Feet  -      Yard 

—  Pole 


40  Poles  .  Rood 

4  Roods,  or  1  60  Rods,  orl  A 

4840  Yards  i         -  Acre 

5.}O  Acres  —  Mile 

so.    SOLID  MEASURE  f 

1728  Inches  make   i  Feet 

27   Feet  Yard 

40  Feet  of  round  timber,  or?  T. 

To  Feet  of  hewn  timber,  Toa  Or  Load 

128  Solid  Feet,  i.e.  8  in  length,  4")  -,     ,    r  TIT 

in  breadth,  and  4  in  height     j  Cord  of  Wood 

*  By  the  calendar,  the  year  is  divided  in  fhe  following  manner  • 

Thirty  days  hath  September,  April,  June,  and  November  ; 

Ttbuary  twenty  eight  alone,  and  all  the  reft  have  thirty  cue. 

When  you  can  divide  the  year  of  our  Lord  by  4,  without  any  re- 
mainder, it  is  then  BiiTextile,  or  Leap  Year,  in  which  February 
has  29  days. 

f  By  Solid  Meafure  are  meafured  all  things  that  have  length, 
breadth,  and  depth, 


2    Pi 


TABLES, 
ii.     WINE  MEASURE.* 


27 


2  Pints 
4  Quirts 
10  Gallons 
1  8  Gallons 
31!  Gallons 
42  Gallons 
Gallons 
Hogftieads 
Pipes 


make  i 


Quart 

G.i;lon 


63 


Anchor  of  Brandy 

Runlet 

Half  HogOiead 

Tierce 

Hogftead 

Pips  or  Butt 

Tun 


marked  pt.   qt. 
gal. 


aac. 
run. 


tier. 

hhd. 

P.  or  B. 

Tun. 


12.     ALE  and  BEER  MEASURE. f 


2  Pints  make   i 

4  Quarts 

8  Gallons 
8*  Gallons 

9  Gallons 
2   Firkins 

2   Kilderkins 

1  \  Barrel,  or  45  Gal. 

2  Barrels 

3  Barrels,  or  2  Hhds. 


Quart  marked         pt.  qt. 

Gallon  gal. 

Firkin  of  Ale  in  London     A.  fir. 

Firkin  of  AL-  or  B^r 

Firkin  of  Beer  in  Lon. 

Kilderkin 

Barrel 

Hogfiiead  of  Beer 

Puncheon 

Butt 


B.fir. 

kil. 

bar* 

hhd. 


pun. 
butt- 


13.     DRY  MEASURE. t. 


2  Pints 

2  Quarts 

1  Pottles 

2  Gallons 
4  Pecks 

2  Bufliels 

2  Strikes 

2  Cooms 

4  Quarters 
^z  Quarters 

5  Quarters 
2  \Vcys 


make  i 


Qujrt 

Pottle 

Gallon 

Peck 

Bulhel 

Strika 

Coom 

Qiurter 

Cii'ildron 

Ch  ildron 

Wcy 

Laft 


marked 


n  London 


pt.  pt. 

pot. 

gal. 

pk. 

bin. 

fir. 

CO. 

qr. 
ch. 

'  wey. 


*  All  Brandies,  Spirits,  Perry,  Cider,  Mead,  Vinegar,  Honey,  and 
Oil,  arc  mcafured  by  Wine  Mc»Unv  ;  H.>acy  is,  co.u.njiiiy,  fold  by 
the  pound  Avoirdupois. 

f   Milk  is  fold  by  ilic  B-cr  quart. 

|  This  meafure  is  apiiLcd  to  all  dry  goods,  as  Corn,  S^cd,  Fruits, 
Roots,  vSalr,  Sand,  Oylkrs,  .OK!  CoaL 

A  Wiachcfttr  hufliclis  iSj;  iuchcj  diaaiccer,  and  8  inches  de:p. 


28  COMPOUND  ADDITION. 


COMPOUND  ADDITION 

IS  the  adding  of  feveral  numbers  together,  having  dif- 
ferent denominations,  as,  Pounds,  (Shillings,  Pence,  &c. 
Tons,  Hundreds,  Quarters,  &c. 

RULE. 

T.  Place  the  numbers,  fo  that  thofe  of  the  fame  de- 
nomination may  ftand  directly  under  each  other. 

2.  Add  the  firft  column  or  denomination  together  as 
in  whole  numbers  ;  then  divide  the  fum  by  as  many  of  the 
fame  denomination,  as  make  one  of  the  next  greater,  fet- 
ting  down  the  remainder  under  the  column  added,  and 
carry  the  quotient  to  the  next  fuperiour  denomination, 
continuing  the  fame  to  the  laft,  which  add  as  in  iimple 
addition. 

EXAMPLES. 

i.    FEDERAL  MONEY. 
i.  2.  3. 

E.  D.  d.    c.  m.  D.     c.    m.  D.     c.     m. 

73895  49     18     7  375 

2125  25     32      i  29     1 8 

9005  93       7     5  7     12     5 

3625  13     25  199     18     7 

71408  97     2  30     01 


tf 

£    s.  d. 

9     16  10 

7     10  9 

o     18  6 

c     ii  ii 


2. 

ENGLISH  MONEY. 

2. 

3- 

£ 

s.   d.  qrs. 

£ 

s. 

d.qrs. 

47 

17 

6 

2 

487 

ii 

Ii 

3 

3 

9 

10 

3 

4i8 

19 

6 

i 

75 

*3 

9 

i 

59 

6 

10 

9 

4 

1  1 

ii 

0 

747 

16 

i 

2 

COMPOUND  ADDITION. 


lb. 

767 

39 

I 
OZ. 
10 

6 

3.  TROY 

2. 

pwt.    lb.  oz. 
17    649  u 

9     32   9 

WEIGHT. 

pwt.  gr. 
19  20 
6   5 

3- 

lb.  oz.  pwt.gr 
859  9  15  20 
437  10  17  22. 

4'7 

1  1 

16    841 

10 

II 

!9 

641 

1  1 

6 

935 

9 

i 

7    473 

9 

17 

23 

n 
i 

38 

9 

12 

18 

4.  AVOIRDUPOIS 

WEIG] 

:IT. 

i. 

2. 

^ 

. 

lb. 

oz. 

dr. 

T.  cwt, 

,  qrs. 

lb. 

T. 

cwt 

.qrs 

.lb. 

OZ. 

dr. 

J9 

13 

12 

59  *3 

2 

'"7 

9' 

17 

2 

2> 

'3 

i-7 

21 

9 

6 

6  17 

I 

2[ 

*9 

9 

0 

17 

10 

4 

*> 

'5 

45  i' 

3 

2? 

14 

13 

2 

O 

9 

1  1 

22 

10 

5 

57  16 

2 

*9 

47 

1  1 

3 

"9 

H 

0 

i 

. 

2 

. 

5 

'3 

gr. 

2 

5 

9 

gr. 

9 

1 

17 

1O 

7 

2 

'9 

3 

2 

'9 

6 

3 

O 

1  2 

6 

I 

^7 

7 

6 

I 

17 

4 

O 

6 

9 

5 

2 

12 

APOTHECARIES'  WEIGHT. 

3- 


lb 

v  —  +<r  i_> 

12  ii  6  i  15 

4  9  i  o  12 

91  10  7  2  16 

4  8  i  2  19 


6.    CLOTH  MEASURE. 

i.  2  3. 

Yd.    qr.  n,  E  £.  qr.  n.  E.F1.  qr.  n. 

76     2     3  9L     3     2  75     2     i 

3     3     x  49     4     3  7i3 

42     33  623  8.1     02 

57     2     2  84    4     i  76     2     3 


C   2 


ADDITION. 


LONG  MEASURE. 
2. 


t      in. 

12        II        1O 

9     10       9 
81211 
7     15       ^ 

Mil.  fur.  pol.         De£.  m.fu  po.  ft.  in.bc. 
9     7     36           759  56  6  29    *5   10  2 
7     3     !9           3*7  39   >  3<5  n     6  i 
4     i     24           497  63  7   24     9     8   i 
6512           562   17011    r  3   i  i  o 

8.    TIME. 

i. 

2. 

5- 

W.d.  h.  m.   s. 

Y.  mo.  d. 

Y.mo.  w.d.  h.  m.    s. 

3  6  22  57  42 

19   10  19 

57   ii   3  6  23  29  55 

i  5   19  31  28 

7     9  27 

4     8   i    i    19  45  38 

2  3   17     9  15 

4     8   16 

29     9  2  3   17   18   19 

30     9  17  58 

j    1  1   14 

46  10  2  5   n  50  13 

9- 

LAND,  OR  SQUARE 

MEASURE. 

i. 

2. 

3- 

Pol.  ft.     in. 

Yd.ft.  in. 

Acr.roo.pol.  ft.     in. 

36  179  137 

28  7   (19 

756  3  37  H5   I28 

19  248   119 

9  3     75 

29   i   28     93     25 

12     96     75 

29  6  120 

416  3  31    128   119 

l8     110    122 

48        12 

37  i    19  218     20. 

i®.  SOLID  MEASURE. 

i. 

2. 

3- 

Ton.  ft.     in. 

Yd.    fc.       in. 

Cord.  ft.        in. 

29     36     12^9 

75     22     1412 

37     119     1015 

12       19            64 

9     26       195 

9     no       159 

l8       II          917 

3     19     *09i 

48     127     1071 

19       8     i  oo  i 

28     15     mo 

8     in       956 

COMPOUND  SUBTRACTION.  31 

ii    WINE  MEASURE. 

i.  2.  3- 

Tie, gal  qt.  pt.       Hhd.gal  qt  pt.         Tun.  hhd.gal.  qt. 

37     39     3     i         51     53     i     i             37     2  37     2 

9     17     2     i         27     39     3     o             19     i  59     i 

4     28     o     o           9     18     o     i              28     2  o     o 

39     19     i      i           o       921             190  47     i 


12.    ALE   AND  BEER  MEASURE. 

i.                             2.  3. 

A.B.fir.gal.  B.B  fir.  gal.  Hhd.  gal.  qt. 

49     3     7  29     i     8  379  53  3 

26     2     3  19     3     5  19  o  i 

904  16     o     3  121  37  2 

17     3     o                       918  467  19  i 


13.    DRY  MEASURE. 

i.  2.  3. 

Qr.  bu.  p.  qt.  Bu.    p.    qt.  pt.  Ch.  bu.  p.  qt. 

64     7     3     7  37     2     5     i  37  27  3     5 

9415  19     370  6  29  17 

19       6       2        I  l6       2       O       I  15  3O  O       O 

4O2O  5l6l  41130 


COMPOUND  SUBTRACTION 

Teaches  to  find  the  difference,  inequality,  or  excefs,  be- 
tween any  two  iums  of  divers  denominations. 

RULE. 

Place  thofe  numbers  under  each   other,  which  are  of 
the  fame  denomination,  the  lefs  being  below  the  greater ; 


32  COMPOUND  SUBTRACTION. 

begin  with  the  leaft  denomination,  and,  if  it  exceed  the  fig- 
ure over  it,  borrow  as  many  units  as  make  one  of  the  next 
greater  ;  fub;ra<5t  it  therefrom  ;  and  to  the  difference  add 
the  upper  figure,  remembering  always,  to  add  one  to  the 
next  fuperiour  denomination,  for  that  wnich  you  borrowed* 


D.      c.    m. 

From  39     15     5 
Take  28     17     2 


D. 

Borrowed    i  oo 
Paid  20 

Remains  to  pay 


EXAMPLES. 

FEDERAL  MONEY. 

E. 

21 
1O 

D.  c.  m.            D.     c.    m, 

812             joo 
75                    48     87     5 

C. 

18 

D.      c.m. 

L?nt    200 
Received  145     50 

Due  to  me 

D.      c. 

Borrowed  3000 


Paid      C  195 

at          J  1 1 15  46 

ieverall    247  37 

times.    C.  995  12 


Paid  in  all 
Remains  to  pay 


D.      c.  m, 

Lent  7159     128 


Received  C 

at  V 

5  feveral  J 

5  times.  (_ 


245  37  5 

3112  15  7 

20CO 

1092  92  o 


Received  in  all 
Remains  due 


2.  ENGLISH  MONEY. 


£.    s.   d.qr. 
Borrowed  349   15  6  i 
Paid  195   ii   8   i 


Remains  to  pay  1^4  3   10  o  Due  me 


£.     li   dq. 

Lent  791     981 
Received  107   16  4  2 


COMPOUND  SUBTRACTION. 


S3 


£.        S.        d.q. 

Borrowed  19372   12     90 


£.        s.    d.q. 

Lent  27109     5     83 


Paid  at 

293 

16 

8 

o 

Rec. 

at 

f 

5196 

'5 

IG 

• 

74 

9 

7 

7 

i 

^84 

16 

6 

2 

fundry 

94  '3 

ii 

0 

i 

feveral 

j 

4187 

18 

1  1 

I 

1994 

0 

10 

3 

6649 

16 

8 

0 

times 

39  '4 

'9 

o 

o 

times 

) 

9*7 

9 

IO 

3 

_  1064 

9 

1 

t 

o 

2 

3 

Paid  in  all 

Rec. 

in  all 

Rem.  to  be  pd. 

Rem 

.  due 

3.    TROY  WEIGHT. 

I.                           2.  3. 

Ib.  oz.  pwt.  gr.       Ib.  oz.  pwt.  gr.  Ib.  oz  pwt.  gr. 

Bought    749  5   13   16     189  8   12   10  543  3     9   13 

Sold           96  9   19   13      148  4   16   19  179   i    15    18 

Rem. 


Bought 
Sold 


4.     AVOIRDUPOIS  WEIGHT. 

I.                             2.  3. 

Ib.    oz.  dr.  C.  qr.  Ib.  T.  c\vt.  qr.  Ib.  oz.  dr. 

7     9   12  8  2   13  9   ii   3    17     5   iz 

3   12     9  4  i    15  3   12   i    19  10     9 


5. 

Ib  §  3  9  gr. 
71  9  3  i  13 
3784116 


APOTHECARIES'  WEIGHT. 


Ib    g5  9  gr. 
65   10  6  2   i 
31     8420 


Ib  3  9  gr/ 
84  i  i  i  i 
6593117 


34  COMPOUND  SUBTRACTION, 

6.     CLOTH  MEASURE. 

i.                      2.                       3  4. 

Yd.  qr    n.        E  E.  qr.  n.         E.FL  qr.  n.  E.Fr.  qr.  n. 

35  i     2       467     3     i       765      i     3  549     4     2 
19     i     3        291      32        '49     2      i  197     4     3 


7.     LONG  MEASURE. 

i.  2  3. 

Yd.    ft      in.  Mil  fur.  pol.          Deg.  m.  fur.  p.  yd.  ft.  in.  bar. 

28  2  10    76  3  ii    38  41  3  29  2172 

17  2  n    27  3  21     19  35  5  31  3  i  9  i 


8.     TIME. 

3- 

Mo.  d.    h.     m.      5,  V.  mo.  d.  Y.  mo.  w.  d.    h.     m.     t , 

6  17   13  27   19         7  3   '3         48  9  *  5   '9  27  3l 
121    i 6  41    3 5  4219  19934201949 


4.  LAND  OR  SQUARE  MEASURE. 

i.  2.                                  3. 

A.   R.   Pol.  A.    R.  Pol.             A.  R.  Pol.    ft.    in. 

29     i      10  29     3      19             56216     27110 

24       I       25  17        I       36                  29    O    21     210    129 


10.  SOLID  MEASURE. 
i.  2.  3. 

T.  ft.  in.      Yds.  ft.   in.      Cords,  it.  in. 
49  19  1100      79  n  917     349  97  125° 
38  36  1296      17  25  1095      192  127  1349 


PROBLEMS.  35 

ii,     WINE  MEASURE. 

i.                              2  3- 

Hhd.gal.    qt.    pr.                  Tier.    gal.    qt.  Tun.  hhd,   gal, 

79    21        2        I                       IQ        17       I  532       I        (9 

38  61     3      i                12     29     2  197     i     47 


12.  ALE  AND  BEER  MEASURE. 

i.  2.                                3. 

A.E.  fir.  gal.    qt.  B.B.  fir.    gal.    qr.     pt.            Hhd.    ga!.     qr, 

39     i      2     i  213520           769     17     i 

24     3     6     2  19     i     7     2     i           39.     42     3 


13,     DRY  MEASURE. 

I.                                               2.  3. 

<>r.    bu.  pk.    qt.              Bu.    pk.     qt.     pt.  Chal.    bu.    pic.    qr. 

56      221               9 1132  39J22I 

39     3     i     2           29     2     i     i  24     25     3     2 


PROBLEMS 

Rs fulling  from  a  Companion  of  the  preceding  Rules. 

PROB.  i.  H  iving  the  fum  of  two  numbers  and  one 
of  them  given  to  find  the  other. 

Rule.  Subtrad  the  given  number  from  the  given  fum, 
and  the  remainder  will  be  the  number  required. 

Let  288  be  the  fum  of  two  numbers,  one  of  which  is 
115,  the  other  is  required. 
From  288  the  fum, 
Take   1 1 5  the  given  number. 

Remains     173  the  other. 

PRCB.  2.  Having  the  greater  of  two  numbers,  and 
the  difference  between  that  and  the  lefs,  given,  to  find  the 
lefs. 


3<5  PROBLEMS. 

Rule.     Subtract  the  one  from  the  other. 
Let  the  greater  number  be  325,  arid  the  difference  be- 
tween that  and  the  other  198  :  What  is  the  other  ? 

From  325  the  gre  rer, 

Take    19^  the  tiifference. 

Rem.     127  the  lefs. 

PROB  3.  Having  the  leaO  of  two  numbers  given,  and 
the  difference  between  that  and  a  greater,  to  find  the 
greater. 

Rule.     Add  them  together. 

Oiven   \  127  the  lefs  number, 
L   1  iy8  the  difference. 

Sum      325  the  greater  number  required. 

PROB.  4  Having  the  fum  and  difference  of  two  num- 
bers given,  to  find  thofe  numbers 

Rule.  To  half  the  fum  add  half  the  difference*  and 
the  fum  is  the  greater  ;  and  from  hall  the  fum  take  half 
the  difference,  and  the  remainder  is  the  lefs. — Or,  From 
the  fum  take  the  difference,  and  half  the  remainder  is  the 
leaft  :  To  the  lealt  add  the  given  difference,  and  the  fum 
is  the  greateft 

What  are  thofe  two  numbers,  whofe  fum  is  48,  and  dif- 
ference 14? 

2)48  2)14 

Half  fum  =  24  Halfdiffrr  7 

24+7=3 1  r^e  greater  ;  and  24 — 7=17  the  lefs. 

Or,  48— -14-4-2  =  17  ;  and  17+14=31. 

PROB.  5  Having  the  fum  of  two  numbers  and  the 
difference  of  their  fquares*  given,  to  find  thofe  numbers. 

Rule.  Divide  the  difference  of  their  fquares  by  the  fum 
of  the  numbers,  and  the  quotient  will  be  their  difference  : 
You  will  then  have  their  fum  and  difference,  to  find  the 
numbers  by  Problem  4. 

What  two  numbers  are  thofe,  whofe  fum  is  32,  and  the 
differ  nee  of  whofe  fquares  is  256  ? 

*  The  fquare  of  a  number  is  the  product  of  it  multiplied  iutc 
itfelf. 


PROBLEMS.  37 

Half  fum  1 6 
Halfdiff.     4 

32)256(8  difference.  — 

256  Greater     20 

Lefs        Tz 

PROB.  6.  Having  the  difference  of  two  numbers,  and 
the  difference  of  their  fquares  given,  to  find  thofe  numbers. 

Rule.  Divide  the  difference  of  their  fquares  by  the  dif- 
ference of  the  numbers,  and  the  quotient  will  be  their 
fum  ;  then  proceed  by  Problem  4. 

What  are  thofe  two  numbers,  whofe  difference  is  2O> 
and  the  difference  of  whofe  fquares  is  2000  ? 

20)2000(100  fum.     50+10=60,  the  greater  ;  and 

50 — 10=40   the  lefs. 

PROB.  7.  Having  the  product  of  two  numbers,  and 
one  of  them  given,  to  find  the  other. 

Rule.  Divide  the  product  by  the  given  number,  and 
the  quotient  will  be  the  number  required. 

Let  the  product  of  two  numbers  be  288,  and  one  of 
them  8  ;  I  demand  the  other.  8)288 

Ans      36 

PROB.  8.  Having  the  dividend  and  quotient,  to  find 
the  divifor. 

Rule.     Divide  the  dividend  by  the  quotient. 
COR    Hence  we  get  another  method  of  proving  divifion. 
r.          ("288  the  dividend,  I       36)188(8    divifor. 

|   36  the  quotient.  288 

Required  the  divifor.  — — 

PROB  9.  Having  the  divifor  and  quotient  given,  to 
find  the  dividend. 

Rule.     Multiply  them  together. 
Given    i    8  the  divifor,  36 

1 36  the  quotient.  8 

Required  the  dividend.  • 

288  the  divdend. 

By  a  due  confideration  and  application  of  the^V  Prob- 
lems only,  many  queitions  may  be  retblved  in  a  fhort  and 
elegant  manner,  although  fome  of  them  are  generally 
fuppofed  to  belong  to  higher  rules. 


3*  APPLICATION  OF  THE 

APPLICATION  OF  THE  PRECEDING  RULES. 

1.  The  lead  of  two  numbers  is  19418,  and  the  differ- 
ence between  them  is  2384:    What  is  the   greater,   and 
fiim  of  both  ? 

19418-1-2384=21802,    greater  ;     and    19418+21802— 
41220,  fum. 

2.  Suppofe  a  man  born  in  the  year  1 743  :  When  will 
he  be  77  years  of  age  ?  17434-77=1820,  Ans. 

3.  What  number  is  that,  which  being  added  to  19418, 
will  make  2*802  ?  21802 — 19418=2384,  Ans. 

4.  General  Wafhington  was  born  in  1732  :    What  was 
his  age  in  1 799  ?  1 799—  1 7  3  2=67,  Ans. 

5.  America  was  difcovered  by  Columbus  in  1492,  and 
its  independence  was  declared  in  1776  :  How  many  years 
elapfed  between  thofe  two  eras  ? 

1776 — 1492=284,  Ans. 

6.  The  maflacre  at  Bofton,  by  the  Britifti  troops,  hap- 
pened March  jth,  1770,  and  the  battle  at  Lexington,  A* 
pril  1 9th,  1775  :  How  long  between? 

April  195  1775 — March  5,  i77o=5y.  im.  i4d.  Ans. 

7.  General  Burgoyne  and  his  army  were  captured  Oct. 
1 7th,  1777,  and  Earl  Cornwallis  and  his  army,  Oct.  i9th, 
178  •  :  What  fpace  of  time  between  ? 

October  19,  1781 — October  17,  1777=47  2d.  Ans. 

8.  The  war  between  America  and  England  commenced 
April  1 9th,  1775,  and  a  general  peace  took  place  January 
2olh,  1783  :  How  long  did  the  war  continue  ? 

Jan  20,  1783 — April  19,  1 775=77-  9m.  id. 

9.  L,  M,  N,  and  O  purchafed  a  quantity  of  goeds  in 
partnerfliip  ;  L  paid  £12   ios?  a  dollar,*  and  a  crownf 
piece  ;  M,  355.  ;  N,  295.  lod.  ;  and  O,  79d.  :  What  did 
the  goods coft  ?  Ans.  £\6  14  i . 

10.  A  man  borrowed,  at  different  times,  thefe  feveral 
fums,viz.  £29  5  ;  £18   17  6  ;   £45   12  ;    £98,  3   dol- 
lars, one  crown  piece,  and  a  half :  How  much  was  he  in 
debt  ?  Ans.  £193  2  6, 


PRECEDING  RULES.  39 

1  1  .  There  are  4  numbers  •,  the  firft  3  1  7,  the  fecoi\d  9  1  3» 
the  third  1229,  and  the  fourth  as  much  as  the  other  three, 
abating  97  ;  What  is  the  fum  of  all  ?  Ans.  4819. 

12.  Bought  a  quantity  of  Goods  for  £1  25  i  os.  paid  for 
truckage  4^3.  for  freight  795.  6d.  for  duties  355.  lod   and 
my  expenfes  were  535.  9<1.  :    What  did  the  goods  Hand 
me  in  ?  Ans.  £136  45.  id. 

13.  A  gentleman  left  his  fon  £1725  more  than  his 
daughter,  whofe  fortune  was  1  5  thoufand  1  5  hundred  and 
1  5  pounds  :  What  was  the  fon's  portion,  and  what  did  the 
whole  eftate  amount  to  ? 

Ans.  The  fon's  fortune,  £18240,  and  the  whole  eftate 


1  4  A  merchant  had  fix  debtors,  who,  together,  owed 
him  £29  1  7105.  6d  A,  B,  C,  D,  and  E,  owed  him  £i6jf 
135.  9d.  of  it  :  What  was  F's  debt  ? 

Ans.  £1241  1  6s.  9d. 

i  5.  What  is  the  difference  between  £1309  75.  id.  and 
the  amount  of  £345  135.  4<d.  and  £$  71  45.  8d.  ? 

Ans.  £392  95.  id. 

1  6  A  merchant,  at  his  firft  engaging  in  trade,  owed 
£937  155.  he  had  in  ca(h£i75$  35.  6d.  in  goods  £459 
i2s  3d.  in  good  debts,  £  197  i6s.  and  he  cleared  the  firft. 
year  £2  49  195.  lod.  What  was  the  neat  balance  at  the 
year's  end?  Ans.  £1724  i6s  7d. 

1  7.  What  fum  of  money  muft  be  divided  between  12 
men,  fo  as  that  each  may  receive 


1  8.  What  number  muft  I  multiply  by  9,  that  the  pr<r- 
dud  may  be  675  ?  175-^-9=7^  Ans. 

19.  A  privateer  of  1  75  men  took  a  prize  which  amount- 
ed  to  £59  per  man,  befides  the  owner's  half:  What  was 
the  value  of  the  prize  ?  1  7  5X5  9X2=^20650  Ans. 

20.  What  is  the  difference  between  thrice  five,  and  thfc- 
ty,  and  thrice  thirty  five  ?         _     —-^  -  - 

35X3  —  5x3+30=60,  Ans. 

21.  The  fum   of  two  numbers  is   750;  the  lefs  248: 
What  istheir  d;fference,  product,  and  the  fquare  of  their  dif- 
ference ? 


40  APPLICATION,  &c. 

750—248=502  the  greater  number,  502  — 248=254  dif- 
ference, 502X2482=! 34496  product,  and  254X354=64516 
fquare  of  the  difference. 

22.  Wh-it  is  the  difference  between  fix  dozen  dozen,  and 
half  a  dozen  dozen  ;  and  what  is  their  product,  and  the* 
quotient  of  the  greater  by  the  lefs  ? 

Ans.  6X12X12—6X12=792  diff. 

6xi  2Xr  2X6X1 2=62208  product,  and  6xi2Xi2-~6x72=i2, 
Quotient. 

23.  There  are  two  numbers  ;  the  greater  of  them  is  25 
times  78,  and  their  difference  is  9  times  15  j  their  fum  and 
product  are  required. 

Ans.  78X25=19^0  the  greater,  1950 — 15X9=1815  the 
lefs.  1 95 0+1815=3 765  the  fum,  and  1950X1815=3539250 
the  product. 

24.  A  merchant  began  trade  with  £25327  ;  for  6  years 
together,  he  cleared  £1253  per  annum  ;  the  next  5  years 
he  cleared  £1729  per  annum  ;  but,  the  laft  4  years,  had 
the  misfortune  to  lofe  £3019  per  annum  :  What  was  he 
worth  at  the  15  years  end  ?  Ans.  £29414. 

25.  If  a  man  fpend£i92  in  a  year,  what  is  that  per 
calendar  month  ;  192-^12=16!.  Ans. 

26.  If  the  Federal  debt,  which  is  42,000,000  dollars, 
be  equally   divided  between   the    thirteen   ftates  ;  what 
will  be  the  fhare  of  each  ?  Ans.  32307695-^013. 

27.  If  9000  men  march  in  a  column   of  750  deep  ; 
how  many  march  abreaft  ?  9000—750=12,  Ans. 

28.  What   number,   deducted  from    the  32d  part  of 
3072,  will  leave  the  96th  part  of  the  fame  ? 

3072-7-32 — 32=64,  Ans. 

29.  What  number  is  that,  which,  being  multiplied  by 
3589,  will  produce  92050672  ? 

92050672-^-3589=25648,  Ans. 

30.  Suppofe  the  quotient  arifing  from  the   divifion  of 
two  numbers  to  be  5379,  and  the  divifor  37625  :  What 
is  the  dividend,  if  the  remainder  come  out  9357  ? 

32625X5379+9357=202394232,  Ans4. 


REDUCTION;  4* 

REDUCTION 

TEACHES  to  bring,  or  exchange,  numbers  of  one  de- 
nomination to  others  of  different  denominations,  retaining, 
tiie  fame  value. 

It  is  of  two  forts,  viz.  Defcending  and  Afcending  ;  the 
former  of  which  is  performed  by  Multiplication,  and  the. 
latter  by  Divifion. 

REDUCTION  DESCENDING. 
RULE. 

Multiply  the  higheft  denomination,  given,  by  fo  many 
of  the  next  lefs,  as  make  one  of  that  greater,  and  thus 
continue,  till  you  have  brought  it  down  as  low  as  your 
queftion  requires. 

PROOF* 

Change  the  order  of  the  queftton,  and  divide  your  laft 
product  by  the  laft  mulciplier,  and  fo  on. 

NOTE.  From  this  rule  and  Cafe  IV.  of  Simple  Multi- 
plication it  appears,  that  Federal  Money  is  reduced  from 
higher  to  lower  denominations  by  annexing  as  many  cy- 
phers as  there  are  places  from  the  denomination  given  to 
to  that  required  ;  or,  if  the  given  fum  be  of  different  de- 
nominations, by  annexing  the  feveral  figures  of  all  the 
denominations  in  their  order,  and  continuing  with  cyphers, 
if  neceffary,  to  the  denomination  required  ;  or,  what  a- 
mounts  to  the  fams  thing,  by  reading  the  wr»ole  number 
from  the  left  to  the  required  denomination,  as  one  num- 
ber in  the  required  denomination. 

EXAMPLES. 

1.  In  3  eagles  2  dollars,  how  many  mills  ? 

Ans.    32ooom. 

2.  In  91  dollars  75  cents,  how  many  cents  ? 

Ans.  91750, 

3.  In  50  eagles,  haw  many  dollars  ?  Ans.  5000. 

4.  In  44.  dollars  i  cent  4  mills,  how  many  mills  ? 

5.  IR  9  dollars  31  cents  7  mills,  how  many  mills  ? 

6.  How  many  cents  in  39  dollars  5  cents  ? 

7.  In  28  dollars  17  ce«nts  5  mills,  how  many  mills  ? 

D  2 


42  REDUCTION. 

8.  In  £27  153.  pd.  2qrs.  how  many  farthings? 
£       s.     d.  qrs. 
27     15     9     2 
Multiplied  by  20  =  (hillings  in  a  pound. 

555  =  Shillings, 
t—  by       12=  pence  in  a  (hilling. 

6669  =  pence. 
.—  »*«_  by         4  =  farthings  in  a  penny. 

Ans.  =  26678  farthings. 


In  multiplying  by  20,  I  added  in  the  155.  ;  by 
I2,the9tl.  ;  and  by  4,  the  2qrs.  which  muft  always  be 
done  in  like  cafes. 

To  prove  the  above  queftion,  change  the  order  of  it, 
audit  will  (land  thus  :  In  26678  farthings,  how  many 
pounds  ? 

4)2667$ 

12)6669     ±qrs. 

2  I  o)55  I  5     9<i- 
Ans.  £27     15     9     2 

9,.  In  £$6  i2s.  lod.  iqr.  how  many  farthings  ? 

Ans.  35177. 
VQ.  In  £95  us.  5d  gqrs.  how  many  farthing^  ? 

Ans.  91751. 
n.  In  £719  95.  i  id.  how  many  half  pence  ? 

An«-  345358. 

12.  In  29  guineas,  at  283.  each,  how  many  pence  ? 

;Ans.  9744. 

13.  In  37  piftoles  at  225.  how  many  (hillings,  pence  and 
farthings  ?  Ans.  8145.  9768d.  39072  farthings. 

14.  In  49  half  j  channes,  at  48^.  how  many  fixpences  ? 

Ans.  4704. 

15.  In  473  French  crowns,  at  6s.  8d.  how  many  three 
pences  ?  An*.  12613-5-. 

16  In  53  moidores,  at  36=.  how  many  (hillings,  pence 
and  Jaj  things  ?  An»,  19082*  22896^. 


REDUCTION, 


43 


17.  In  Lig  how  many  groats,  threepences,  pence  and 
farthings  ? 

Ans.  1740  groats,  2320  threepences,  6$6od.  27840915, 

1  8.  Reduce  47  guineas  and  one  fourth  of  a  guinea  into 
(hillings,  fixpences,  groats,  three  psnces,  twopences,-  pence 
and  farthings. 

Ans.  1323  (hillings,  2646  fjx-pences,  3969  groats,  5292 
three-pences,  7938  two-pences,  158764.  and 


REDUCTION  ASCENDING. 

RULE. 

Divide  the  lowed  denomination  given,  by  fo  many  of 
that  name,  as  make  one  of  the  next  higher,  and  thus  con- 
tinue, till  you  have  brought  it  into  that  denomination, 
which  your  queftion  requires. 

NOTE.  From  this  rule  and  the  note  under  Cafe  II.  of 
Simple  Divifion,  it  appears,  that  Federal  Money  is  redu- 
ced from  lower  to  higher  denominations  by  cutting  off  as 
many  places  as  the  given  denomination  (lands  to  the  right 
of  that  required  ;  the  figures  cut  off  belonging  to  thur 
refpeclive  denominations. 

EXAMPLES. 

i.  How  many  eagles  in  32000  mills  ?     Ans.  3E.  ?D. 
"2.  In  9175  cents,  how  many  dollars  ?  Ans.  9iD.  750. 

3.  In  500  dollars,  how  many  eagles  ?  Ans.  50. 

4.  In  4414  mills,  how  many  dimes  ? 

5.  In  9317  mills,  how  many   dollars  ? 

6.  How  many  dollars  in  28175  mills  ? 

7.  In  547325  farthings,  how    many  pence,   (hillings, 
and  pounds  ? 

Farthings  in  a  pennny       =:     4)5473  *5 

Pence  in  a  (hilling  =   12)136831      iqr. 

Shillings  in  a  pound          =    2^0)1  140]  2     7d. 

£570  2s.   7<L    iqr, 
Ans.  1  368  j  id.  ;  114025.  ;  5^0!. 


4i  REDUCTION. 

NOTE.     The  remainder  is  always  of  the  fame  name. as 
the  dividend. 

8.  Bring  35177  farthings  into  pounds. 

9.  Bring  91751  farthings  into  pence,  &e. 

10.  Bring  345358  half  pence  into  pence,  (hillings,  and 
pounds. 

11.  Reduce    9744   pence    into  guineas,    at    283.   per 
guinea. 

12.  In  39072  farthings,  how  many  piftoles,  at  22s.  ? 

13.  In  4704  fix-petices,  how  many  half  Johannes  ? 

14.  In  126133- three-pences,  how  many  French  crowns, 
at  6s.  8d. 

15.  In  91584  farthings,  how  many  moidores,  at  363.  ; 

1 6.  In  27840  farthings,  how  many  pence,  three-pences, 
groats,  (hillings,  and  pounds  ? 

17.  In  63504  farthings,  how  many  pence,  two-pences, 
three-pences,  groats,  fix  pences,  (hillings,  and  guineas  ? 

NOTE.     The  preceding  queftions  may  ferve  as  proofs 
to  thofe  in  Reduction  defcending. 


REDUCTION  DESCENDING  and  4SCEND1NG. 

i.  MONEY. 

1.  In  ^97   how  many  pence,  and  English  or  French 
crowns,  at  6s.  8d.  ?  Ans.  2328od.  and  291  crowns. 

2.  In  5 1 9   Englifti  half  crowns,  how  many  pence  and 
pounds?  Ans.  2076cd.  and  861.  los. 

3.  In  735    French  crowns,   how    many  (hillings  and 
French  guineas,  at  265.  8d. 

Ans.  49005.  and  183  guineas,  2os. 

4.  In  5793  pence,  how  many  farthings,  pounds  and 
piltoles  ? 

Ans.  23i72qrs. ;  241.  2s.  9d.  ;  and  21  piftoles,  2cs.  9d. 

5.  In  59  half  joes,   37  moidores,  45  guineas,  63  pift- 
oles, 24  Englifh  crowns,  and   19  dollars  j    how  many 


REDUCTION. 


45 


pounds,  half  joes,  moidores,  guineas,  piftoles,  Englifli 
crowns,  dollars,  (hillings,  pence  and  farthings  ? 

Ans.  354.1.  45.  ;  147  half  j  ics,  28$.  ;  196  moidores, 
28s.  j  253  guineas;  322  piftoles;  1062  Englifh  crowns, 
45.  ;  1 1 80  dollars,  43. ;  7084  ihillings ;  85008  pence,  and 
340032  farthings. 

When  it  is  required  to  know  how  many  forts  of  coin, 
of  different  values,  and  of  equal  number,  are  contained 
in  any  number  of  another  kind  ;  reduce  the  feveral  forts 
of  coin  into  the  loweft  denomination  mentioned,  and  add 
them  together  for  a  divifor  ;  then  reduce  the  money  given 
into  the  fame  denomination  for  a  dividend,  and  the  quo- 
tient arifmg  from  the  divifion  will  be  the  number  required. 

NOTE.  Obferve  the  fame  direction  in  weights  and 
meafures. 

i.  In  275  half  Johannes,  how  many  moidores,  guineas, 
piftoles,  dollars,  (hilling?,  and  fix-pences,  of  each  the  like 
number  ? 

A  moid,  is  363.  =  72  fixpences.  275  half  joes. 

A  guinea  is  285.=  56  do.  483.  in  a  johan. 

A  piftole  is  22  s.  =  44  do.  — 

A  dollar  is  6s.    =12  do.  2200 

One  {hilling        =    2  do.  1100 

i  do.  ^ 

13200    (hillings. 

Divifor  —   187  fixpences.  2  fix-p.  in  a  fhill. 


Dividend  =  26400  fixpences. 

187)26400(141    of  each,  and  33  fix-pences,  or   i6s.  6d. 
over,  the  anfwer. 

2.  A  gentleman  diftributed  37!.  IDS.  between  4perfons 
in  the  following  manner,  viz.  that  as  often  as  the  firfthad 
2os.  the  fecond  fhould  have  155.  the  third  los.  and  the 
fourth  53  What  did  each  perfon  receive  ? 

Ans.  The  firft  man  15!.  ;  fecond  7!.  ics.  j  third  3!.  i^s. 

2.    TROT  WEIGHT. 

i.  How  many  grains  in  a  filverbowl,  that  weighs  3lb. 
10  oz.  i  zpwt.  ? 


4<»  REDUCTION, 

lb.     oz.  pwt. 

3       10       12 

1 2  ounces  in  a  pound. 

46  ounces. 

20  pennyweights  in  an  omitis.    . 

932  pennyweights. 
24  grains  in  on*  pwt. 

3728 
1864 

Proof    24)22*68  grains,  Ans, 
2  1  o;93U 

12)46       I2pwt. 

3lb.   looz. 

2.  In  13  ingots  of  gold,  each  weighing  902  5pwt.  how 
many  grain  ?  Ans.  5772cgr. 

3.  In  97397    grains,  how  many  pounds  ? 

Ans.   i61b.   looz.    iSpwt.  5gr. 

4.  How  many  rings,  each  weighing  5 pwt.  7gr.  may  be 
made  of  3\b.  5oz.  i6pwt.  2gr.  of  gold  ?          Ans.   15$, 

3.     AVOIRDUPOIS  WEIGHT. 

Cwt.  qrs.  ib.     oz. 
i.    In  91     3     17     14   how  many  ounces  ? 

4 

367  quarters.  Proof. 

28  16)164702 

2943  28)10293    I4d2.» 

4)367     I7lb. 
10293  pounds.  — — 

-JLl  Gwt>  9l 

61762 
10293 

164792  ounces. 


REDUCTION. 


47 


'2.  In  12  tons,  ijcwt.  iqr.  i9lb.  6oz    13  dr.  how  many 
drams?  Ans.  7323500^. 

3.  In  24!!).  i  loz.  pdr.  how  many  drams  ? 

Ans.  632pdr. 

4.  In   44800  pounds,  how  many  drams  and  tons  ? 

Ans.   ii4688ocdr.  and  20  tons. 

5.  In  28  pounds  avoirdupois,  how  many  pounds  Troy  ? 

281b. 
7000  grains  in  lib.  avoir. 

grs,  in  lib.  Tr.  =576!o)i96oo|o(34lb. 
1728 

2320 
2304 

1  60 

12 

57610)19210(0  oz. 
20 


3456 

3840 
24 


768 

576(0)92  1  6|o(  1  6gr. 

576 


345^ 


48  REDUCTION. 

6.  In    47lb.    907.   ijpwt.    I7grs.    Troy,   how  many 
pounds  avoirdupois  ? 

Ib.    oz.  pwt.  gr. 
47     9     J3     17 

12 

573 

20 

1H73 
24 

45899 
22947 

7(000)2751369(39^. 

2X 
65 


2369 

16 

14214 

2369 

71000)371904(502. 
35 


7)000)461464(6^5  drams- 
42 

4464 


REDUCTION. 

4.     APOTHECARIES'  WEIGHT. 

i.  How  many  grains  are  there  in  37lb.  63;  ? 

lb.     §  Proof. 

37     6  2Jo)2i6oc|o 

1  2  ---- 

-  -  3)10800 

450  ounces.  •*  -  —  - 

8  8)3600 

3600  drams.  12)450 

-  - 
10800  fcruples. 
20 


2  J  6000  grains. 

2.  In9lb.  8  1  13  2  5   iggr.  how  many  grains  ? 

Ans. 

3.  In  55  7  99  grains,  how  many  pounds,  &c.  ? 

Ans-  9lb.  8oz.   15  2  9 

5.    CLOTH  MEASURE. 

I.  In  127  yards,  how  many  quarters  and  nails  ? 
Yds. 
127  Proof. 

4 


508  qrs.  4)508 

—  —  127  yds. 

Ans.  2032  nails. 

2.  In  9173  nails,  how  many  yards  ? 

Ans.  573yds.  iqr.   in. 

3.  In  75  ells  Englifh,  how  many  quarters  and  nails  £ 

Ans.  375qrs.  150011. 

4.  In  56  ells  Flemifh,  how  many  quarters  and  nails  ? 

Ans.   i68qrs.  67211.' 

e.  In  7248  nails,  how  many  yards,  ells  Flemifb,  and 
ells  Englifh  ? 

Ans.  453yds,  604  ells  Flemifh,  362  ells  En,  zqrs, 

E 


REDUCTION. 


6.  In   1 9  pieces  of  cloth,  each  ijyds.  2qrs.  how  many 
yards,  quarters,  and  nails  ? 

Ails.  294yds.  2qrs.  ;  H78qrs.  and  47i2n. 

6.    LONG  MEASURE. 

i .  How  many  barley-corns  will  reach  from  Newbury- 
port  to  Bofton,  it  being  43  miles  ? 
Miles. 


43 
8 

344  furlongs. 
40 

13760  rods. 


68800 
6880 

75680  yards, 
3 

227040  feet. 

12 


Proof. 

3)8.73440 

12)2724480 
3)227040 
11)75680 


2724480  inches. 
3 


.8173440  Ans. 

Here  1  divide  by  1 1  and  multiply  the  quotient  by  2, 
becaufe  twice  5*  is  n  ; — or,  I  might  firft  have  multiplied 
by  2,  and  then  have  divided  the  product  by  1 1. 

2.  How  many  barley-corns  will  reach  round  the  globe, 
it. being  360  degrees?  Ans.  4755801600. 

3.  How  many  inches  from  Nswburyport  to  London, 
it  being  * 7co  miles  ?  Ans.   171072000. 

4.  How  often  will  a   wheel,   of  16  feet   and  6  inches 
circumference,  turn  round  in  the  diftance  from  Newbury- 
port  to  Cambridge,  it  being  42  miles  ? 

Ans.  13440  times. 


REDUCTION. 


5.  In    190080  inches,  how  many  yards  and  leagues  ? 
Ans.  528oyds.  and  i  league. 


7.    TIME. 

i.  In  20  years,  how  many  feconds 

d.     h. 

365     6  in  a  year. 
24 

1466 
73° 

8766  hours  in  one  year. 
20 

175320  hours  in  20  years, 
60 


Proof. 
6(6)631 1 5200(0 


10519200  minutes  in  do. 
60 


63115  zooo  feconds  in  do. 

2.  Suppofe  your  age  to  be  ijy.   I9d.  nh.  37m.  455. 
how  many  feconds  are  there  in  it,  allowing    365  days  6 
hours  to  the  year  ?  Ans.  475047465. 

3.  How  many  minutes  from  the  fir  (I  day  of  January 
to  the  1 4th  day  of  Auguft,  inciufively  ? 

Ans.    325440. 

4.  How  many  days  fmce   the  commencement  of  the 
Chriftian  era  ? 

5.  'How  many  minutes  fince  the  commencement  of  the 
American  war,  which  happened  on  the  iQth  day  of  A- 
P"li  '775  - 

6.  How  many  feconds  between  the  commencement  of 
the  war,  April  I9th,  1775,  and  the   Independence  of  the 
United  States  of  America,  which  took  place  the  4th  day 
of  July,  1776  ?*  Ans.  3-8188800. 

*  1776  was  a  JLeap  Y£ar, 


52  REDUCTION. 

8.    LAND,  OR  SQUARE  MEASURE. 

1.  In  29  acres,  3  roods,  19  poles,  how  many  roods  and 
perches  ? 

Acr.     r.     p.  Proof. 

29     3     19  4|°)477l9 

4)119    19?. 

i  \  9  «roods»  - — 

40  293.  $r. 

Ans.  4779  perches. 

2.  In  1997  poles,  how  many  acres  ? 

Ans.  i2a.  ir.  37p. 

3.  In  89763  fquare  yards,  how  many  acres,  &c.  ? 

Ans.  i8a.    2r.  7 p.   joift.   3j5in. 


9.    SOLID  MEASURE. 

1 .  In  i  c  tons  of  hewn  timber,  how  many  folid  inches  ? 

15  tons.  Proof. 

50  5|o 

1728)1296000(75)0 

750  feet.  12096      — 

1728  . 15  To. 

8640 

6000  8640 

1500 
5250 
750 

1 296000  inches,  Ans. 

2.  In  9  tons  of  round  timber,  how  many  inches  ? 

Ans.  622080* 

3.   In  25  cords  of  wood,  how  many  inches  ? 

Ans.  5529600*. 


REDUCTION.  *3 

10.  WINE  MEASURE. 

1.  In  Qhhd,  if  gal.  3qt.  of  wine,  how  many  quarts  ? 

hhd.  gal.  qt. 

g     1S     3  Proof. 

63  4)'33' 

~  63)582       3qtS. 

,.     -  9-hhd.    1 5galv 

582  gallons. 
4 

2331  quarts,  Ans. 

2.  In  12  pipes  of  wine,  how  many  pints  ? 

Ans.  1209(5. 

*.  In  0758  pints  of  brandy,  how  many  pipes  ? 
Ans.  9?.    i  hhd.  2 /gal. 


ii.     ALE   AND  BEER  MEASURE. 

i.  In  29hhd.  of  beer,  how  many  pints  I 
hhd. 

29  Proof. 

j  2)12528 


1566  gallons. 


6264  quarts. 

2 

12528  pints. 
2.  In  47  barrels,  i8gal.  of  ale,  how 

?.  In  a6  puncheons  of  beer,  how  many  btitu  ? 

Ans.  24, 

E  4 


54  VULGAR  FRACTIONS. 

12.    DRY  MEASURE. 

r.  In  42  chaldrons  of  coals,  how  many  pecks  & 
Chaldrons. 

42  Proof. 

52  4)5376 

84  32-)1 344(4a 

126  128 

1344  buftiels.  64 

4  64 

Ans.  5376  pecks. 

2*  In  75  bufliels  of  corn,  how  many  pints  ? 

Ans.  4800. 

34,  In  9376  quarts,  how  many  bufliels  ?  Ans.  293. 


VULGAR  FRACTIONS. 

FRACTIONS,  or  broken  numbers,  are  expreflions  for 
any  affignable  parts  of  a  unit,  or  whole  number  ;  and 
are  represented  by  two  numbers,  placed  one  above  anoth- 
er, with  a  line  drawn  between  them,  thus,  •£,  4*  &c.  fig- 
nifying  five  eighths,  four  thirds,  that  is,  one  and  one 
third,  &c. 

The  figure  above  the  line  is  called  the  numerator,  and 
that  below  it,  the  denominator. 

The  denominator  (which  is  the  divifor  in  divifion) 
fnows  how  many  parts  the  integer  is  divided  into  ;  a^d  the 
numerator  (which  is  the  remainder  after  divifion)  fhows 
how  many  of  thofe  parts  are  meant  by  the  fraction. 

Fractions  are  either  proper,  improper,  fingle,  com- 
pound, or  mixed.  Any  whole  number  may  be  made  an 
improper  fraction,  by  drawing  a  line  under  it,  and  put- 
ling  unity,  or  i ,  for  a  denominator,  as  9  may  be  exprefled 
fraction  wife,  thus,  £,  and  12  thus,  *T*,  &c. 

i.  A  Jingle  or  fimfle  fraction  is  a  fraction  esrpreflfed  if? 
form  ;  as,  4,  f,  T7T,  &c.- 


VULGAR  FRACTIONS.  55 

3.  A  compound  fraction  is  a  fraction  exprefled  in  a  com- 
pound form,  being  a  fraction  of  a  fraction  ;  or  two  or 
more  fractions  connected  together  ;  as,  ^  of  ^,  4  of  ^ 
of  4§  »  which  are  read  thus,  one  half  of  three  fourths, 
two  fevenths  of  five  elevenths  of  nineteen  twentieths,  &c. 

3.  A  proper   fraction  is  a  fraction,  whofe  numerator  is 
lefs  than  its  denominator  ;  as,  f ,  -J-,  &c. 

4.  An  improper  fraction  is  a  fraction,  whofe  numerator 
exceeds  its  denominator  ;  as,  ^,  4>  &c« 

5.  A  mixed  number   is  compofed   of  a  whole  number 
and  a  fraction  j  as,  7]-,  35/7,  Sec.  that  is,  feven  and  three 
fifths,  £c. 

6.  A  fraction  is  faid  to   be  in   its   leaft   or  loweft   ter?nsy 
when  it  is  exprefled  by  the  leall  numbers  poflible. 

7.  The    common   meafare  of  two,  or   more  numbers,  is 
that  number,  which  will  divide  each  of  them   without  a 
remainder  :  Thus,  5  is  the  common  meafure  of  10,  20, 
and  30  ;  and  the  greatefl  number,   which  will  do  this,  is 
called  the  greateft  common  meafure. 

8.  A  number,  which  can  be  meafmed  by  two,  or  more, 
numbers,  is  called  their  common  multiple  :  And,  if  it  be  the 
leall  number,  which  can  be  fo  meafured,  it  is  called  the 
leaft  common  multiple  ;  thus,  40,  60,  80,  100,  are  multiples 
of  4  and  5  ;   but  their  leaft  common  multiple  is  20. 

9.  A  prime  number  is  that,  which  can  only  be  meafured 
by  itfelf,  or  an  unit. 

10.  That   number,  which  is  produced  by  multiplying 
feveral  numbers  together,  is  called  a  compoftte  number. 

1 1 .  A  perfeft  number  is  equal  to  the  fum  of  all  its  all- 
quot  parts. 

PROBLEM    I. 
To  Jlnd  the  great  eft  common  meafure  of  tiuo  or  mors  numbers* 

RULE. 

T.  If  there  be  two  numbers  only,  divide  the  greater  by 
the  lefs,  and  this  divifor  by  the  remainder,  and  fo  on,  al- 
ways dividing  the  latt  divifor  by  the  Jaft  remainder,  till 
nothing  rem  an,  then  will  the  lail  divifor  be  the  greatefl 
common  meafure  required. 

2.  Wrien  there  are  more  than  two  numbers,  find  the 
greatest  common  meafure  of  two  of  them,  as  before  j 


$6  VULGAR  FRACTIONS. 

then  of  that  common  meafure  and  one  of  the  other  num- 
bers, and  fo  on,  through  all  the  numbers  to  the  laft  j 
then  will  the  greateft  common  meafure,  laft  found,  be  the 
anfwer. 

3.  If  i  happens  to  be  the  common  meafure,  the  given 
numbers  are  prime  to  each  other,  and  found  to  be  incom- 
menfurablc,  or  in  their  loweft  terms. 

EXAMPLES. 

i.  What  is  the  greateft  common  meafure  of  1836, 
3996,  and  1044  ? 

1836)3996(2  So  i c8  is  the  greateft  common 

3672  meafure  of  3996  &  1836  5 

Hence  108)1044(9 

324)1836(5  972 

1620  — 

72)108(1 

216)324(1  72 

2.16  — 

Laft  gr.  com.meaf.=36)72(2 

Com.  meaf.=io8)2 16(2  72 

216  — 

Therefore  36  is  the  anfwer  required. 

2.  What  is  the  greateft  common  meafure  of  1224  and 
1080  ?  Ans.  72. 

PROBLEM   II. 

To  find  the  leaft  common  multiple  of  two,  or  more,  nwnbers. 

1.  Divide  by  any  number,  that  will  divide  two  or  more 
of  the  given  numbers,  without  a  remainder,  and  fct  the 
quotients,  together  with  the  undivided  numbers,  in  a  line 
beneath. 

2.  Divide  the  fecond  line  as  before,  and  fo  on,  till  there 
are  no  two  numbers,  that  can  be  divided  ;  then,  the   con- 
tinued product  ©f  the  divifors  and  quotients  will  give  the 
multiple  required. 

EXAMPLES. 

T.  What  is  the  leaft  common  multiple  of  6,   10,  i6> 
and  20  ? 


VULGAR  FRACTIONS.  57 

*5)6     10     16     20 
*2)6       2     16      4 


*3 •     *4       ' 

*    #     *     #     # 

5X2X2X3X4^=240  Ans. 

I  furvey  my  given  numbers,  and  find  that  5  will  divide 
two  of  them,  viz.  10  and  20.  which  I  divide  by  5,  bring- 
ing, into  a  line  with  the  quotients,  the  numbers,  which  5 
wul  not  meafure  :  again,  I  view  the  numbers  in  the  fee- 
on^  line,  and  find  2  will  meafure  them  all,  and  I  get  3,  I, 
8,  2,  in  the  third  line,  and  find  that  2  will  meafure  8  and 
2,  and  in  the  fourth  line  get  3,  1,4,  i,  all  prime;  I  then 
muhiply  ti  e  prime  numbers  and  the  divifors  continually 
into  each  oilier,  for  the  number  fought,  and  find  it  to  be 
240. 

2.  What  is  the  leaft  common  multiple  of  6  and  8  ? 

Ans.  24, 

3.  What  is  the  leaft  number  that  3,  5,  8,  and   to  will 
meafure?  Ans.    120. 

4.  What  is  the  leaft  number,  which  can  be  divided  by 
the  9  digits  feparately,  with  out,  a  remainder  ? 

Ans.  2520. 

REDUCTION  OF  VULGAR  FRACTIONS 

Is  the  bringing  of  them  out  of  one  form  into  another, 
in  order  to  prepare  them  for  the  operations  of  Addition, 
Subtraction,  &c. 

CASE  I. 

To  abbreviate,  or  reduce  frattions  to  tke'tr  loweft  terms. 

RULE.* 

Divide  the  terms  of  the  given  fraclion  by  any  number, 
which  will  divide  them  without  a  remainder,  and  the  quo- 

*  That  dividing  both  the  terms,  that  is,  both  numerator  and  de- 
nominator of  the  fractions,  equally  by  any  number,  whatever,  will 
give  another  fraction,  equal  to  the  icimer,  evident :  And  if  thofe 


58  REDUCTION  OF 

tients  again  in  the  fame  manner  ;  and  fo  on,  till  ft  ap- 
pears that  there  is  no  number  greater  than  i,  which  will 
divide  them,  and  the  fraction  will  be  in  its  loweft  terms. 

Or,  Divide  both  the  terms  of  the  fraction  by  their  great- 
eft  common  meafure,  and  the  quotients  will  be  the  terms 
of  the  fraction  required. 

EXAMPLES. 
i.  Reduce  |££  to  its  loweft  terms: 

-i  (4)     (?)   I 

Bjltt  =  l4=T9r  <=  f  the  anfwer. 

Or  thus  : 

28-8)480(1  Therefore  96  is  the  greateft  com- 

288  meafure  ; 

ToT)  z88(  i  and  96  }  ***=*  the  fame  as  before: 


ToT)  z88( 
192 

Com.  mea.  96)  192  (2 
192 

2.  Reduce  7*TV  to  »ts  loweft  terms.  Ans.  |. 

3.  Reduce  //T  to  its  loweft  terms.  Ans.  •£. 

4.  Reduce  4-|f  |  to  its  loweft  terms.  Ans.  f. 


divlfions  be  performed  as  often  as  can  be  done,  or  the  common  di- 
vifor  be  the  greateft  poffihle,  the  terms  of  the  refulting  fradlion 
muft  be  the  lead  poflible 

NOTE  r.  When  numbers  with  the  fign  of  Addition  or  Sub- 
tradlion  between  them  are  to  be  divided  by  any  numbers,  each  of 
the  numbers  mufl  be  divided:  Thus,  6-f-9-|-ia=2-f-3-f*=9. 

3 

a.  But  if  the  numbers  have  the  fign  of  Multiplication  be- 
tween them  ;  then  only  one  of  them  muft  be  divided  :  Thus; 


1X5 


VULGAR  FRACTIONS.  59 

CASE  II. 
To  reduce  a  mixed  number  to  its  equivalent  improper  fraftion* 

RULE* 

Multiply  the  whole  number  by  the  denominator  of  the 
fraction,  and  add  the  numerator  of  the  fraction  to  the  pro- 
duct ;  under  which  fuHjoin  the  denominator,  and  it  will 
form  the  fraction  required 

EXAMPLES. 
i.  Reduce  36!  to  its  equivalent  improper  fraction. 

36  I  multiply  3.6  by  8,  and  adding  the 

X8+5  numerator  5  to  the  product,  as  I  mul- 

tiply, the  fum,  293.   is  the  numerator 

Ans.  293  of  the  fraction  fought,  and  8  the  de- 

-  nominator  :  So    that  ^      is  the    im- 

proper traction,  equal  to 


Or    fS  ~  223   Ans.  as  before. 
8  8 

2.  Reduce  127/7  to  its  equivalent  improper  fraction. 

Ans. 


3.  Reduce  653rV  to  its  equivalent  improper  fraction. 

A  __          1      4  1  O 

Ans.   —75-. 

CASE     III. 

70  reduce  a  whole  number  to  an  equivalent  fraflhn9  having  tt 
given   denominator. 

RULE. 

Multiply  the  whole  number  by  the  given  denominator  : 
Place  the  product  over  the  faid  denominator,  and  it  will 
form  the  fraction  required 

EXAMPLES. 

I.  Reduce  6to  a  fraction,  whofe  denominator  fhall  be  £• 
6x8=48,  and  V  the  Ans.—  Proof. 


2    Reduce  15  to    a   fraction,    whofe  denominator  fiiall 
be  12.  Ans.  ±5£. 

*  All  fractions  reprefcnt  a    divifion   of  a  numerator  by  the  de- 
nominator, and    are    taken  altogether  a&   proper   and  adequate  »:x- 
the  quotient.    Thus  the  quotient  of  jf  divided   by  4, 


oo  REDUCTION  OF 

CASE   IV.* 

To  reduve  an  improper  frattion   to  its  equivalent  whole  of 
mixed  number. 

RULE. 

Divide  the  numerator  by  the  denominator  ;  the  quo- 
tient will  be  the  whole  number,  and  the  remainder,  if  any, 
will  be  the  numerator  to  the  given  denominator. 

EXAMPLES. 

i  .  Reduce  -|-2  to  its-  equivalent  whole  or  mixed  num- 
ber. 


24 

53 
48 

—  Or,  ±|J=293-j-8=r36|-  as  before. 

5 

2,  Reduce  iil^  to  its  equivalent  whole  or  mixed  num- 
ber. Ans.    i27rV 

3.  Reduce  y  to  its  equivalent  whole  number. 

Ans.  9. 

CASE    V. 

¥0  reduce  a  compound  frattion  to  an  equivalent  finiple  one. 

RULE. 

Multiply  all  the  numerators  continually  together  for  a 
hew  numerator,  and  all  the  denominators  for  a  new 
denominator,  and  they  will  form  the  limple  fraction  re- 
quired. 

If  part  of  the  compound  fraction  be  a  whole  or  mixed 
number,  it  muft  be  reduced  to  an  improper  fraction,  by 
caie  2  dor  3d. 

If  the  denominator  of  any  member  of  a  compound 
fraction  be  equal  to  the  numerator  of  another  member 
thereof,  thefe  equal  numerators  and  denominators  may- 
be expunged,  and  the  other  members  continually  multi- 

*  ThUcare  is  evidently  the  reverfe  of  Cafe^d. 


VULGAR  FR  ACTIONS.  tfi 

plied,  (as  by  the  rule)  will  produce  thefraaions  required 
in  lower  terms. 

EXAMPLES. 

i  .  Reduce  ^  of  f  of  |  of  4  to  a  fimple  fraction. 
1x2x3x4 
2X3X4X5 

Or,  by  expunging  the  equal  numerators  and  denomi- 
nators, it  will  give  3  as  before. 

2.  Reduce  J  of  £  of  -J-  of  -\-^  to  a  fimple  fraction. 
3X4X5X1  '=1^=^  Ans.     Or,  by  expunging  the  equal 

numerators  and  denominators,  it  will  be  3Xf  *  =  ||  =  ££ 

OX  1  2 

as  before. 

3.  Reduce  £  of  -J  of  J  of  12^  to  a  fimple  fradion. 

Ans.  #.!«. 

CASE   VI. 

5T0   reduce  fraftions  of  different   denominators  tv  equivalent 
JraSwnsy  having  a  common  denominator. 

RULE  I. 

Multiply  each  numerator  into  all  the  denominators, 
except  its  own,  for  a  new  numerator,  and  all  the  denomi- 
nators into  each  other  continually,  for  a  common  de- 
nominator. 

EXAMPLES. 

1.  Reduce  ^,  J,  and  J  to  equivalent  fractions,  having 
a  common  denominator. 

1x5x8=  40  the  new  numerator  for  ^. 
2X4X8=  64  ditto  £. 

5X4X5=100  ditto  4-. 

4X5X8=160  the  common  denominator. 

Therefore  the  new  equivalent  fractions  are  y^,  y^j, 
and  T§£»  the  anfwer. 

2.  Reduce  J,  |,  |,  |,  and   |,  to  fra&ions,  having  a 
common  denominator. 

Anc        576          -J6B          86*          960        1008 
"•»»• 


11  JJ>   TI72>  TTTi?' 


REDUCTION  OF 
3.  Reduce  -J-,  |  of  £,  7|,  and  T3T  to  a  common  dencrm- 

936         1O4O       145O8         432 


RULE  II. 

To  reduce  any  given  frafiions  to  others,  which  fuall  have  the 
leaft   common  denominator. 

1.  By   Problem  2,  (page   56)  find  the  leaft  common 
multiple  of  all  the  denominators  of  the  given  fractions, 
and  it  will  be  the  common  denominator  required. 

2.  Divide  the  common  denominator  by  the  denomina- 
tor  oi  each  fraction,  and  multiply    the    quotient  by  the 
numerator,  and  the  product  will  be  the  numerator  of  the 
fraction  required. 

EXAMPLES. 

i.  Reduce  ^,  |,  and  ^  to  fractions  having  the  leaft  com- 
mon denominator  poffible. 

4)3     4     8 

24=lealt  common 
denominator. 


24 -1.3X1  =8  the  firft  numerator  ;  24—4X3=18  the  fecond 

numerator  ;  24-^8X7=21  the  third  numerator. 

Whence  the  required  fractions  are  ^  ^,  |^. 

2.  Reduce  ~,  |,  |,  and  -J  to  fractions  having  the  Jeaft 
common  denominator.  Ans.  |§,  -J-J,  |^,  J.J. 


CASE    VII. 

*£Q  reduce  a  fraction  of  one  denomination   to  the  fraction  of 
another,  but  greater^  retaining  the  fame  "value. 

RULE. 

Reduce  the  given  fraction  to  a  compound  one  by  com- 
paring it  with  all  the  denominations  between  it  and  that 
denomination  you  would  reduce  it  to  ;  laftly,  reduce  this 
compound  fraction  to  a  fingle  one,  by  Cafe  V  and  you 
will  have  a  fraction  of  the  required  denomination,  equal 
j&  value  to  the  given  fraction* 


VULGAR  FRACTIONS, 


EXAMPLES. 

. .  Reduce  4  of  a  cent  to  the  fraclion  of  a  dollar. 
By  comparing  it   it  becomes  4  °f  T<T  °f  Tff>  which  redu- 
ced by  Cafe  V.  will  be  4XjJ<  <_==  JL_  =    ^  QoL  Ans> 

and  7X10X10=  700 
2.   Reduce  ^  of  a  penny  to  the  fraction  of  a  pound. 


3.  Reduce   4  of  an  ounce  to  the  fraction  of  a  Ib   Av- 
oirdupois. Ans.  ^-glb. 

4.  Reduce  -J-  of  a  pennyweight  to  the  fraction  of  a  Ib. 
Troy.  Ans.  r^Vo-lb. 

5.  Reduce  |  of  a  Ib.  Avoirdupois  to  the  fraction  of  a 
Gwt.  Ans. 


CASE     VIII. 

To  reduce  a  fraction  of  one  denomination  to  the  frattion  of  CQ- 
cther^  but  /efs,  retaining   the  fame  value. 

RULE. 

Multiply  the  given  numerator  by  the  parts  of  the  de- 
nominations between  it  and  that  denomination  you  would 
reduce  it  to,  for  a  new  numerator,  which  place  over  the 
given  denominator  :  Or,  only  invert  the  parts  contained 
in  the  integer,  and  make  of  them  a  compound  fraftion  as 
before  ;  then,  reduce  it  to  a  fimple  one. 

EXAMPLES. 

1.  Reduce  y-^  of  a  dollar  to  the  fraction  of  a  cent. 
By  comparing   the   fraction    it  will  be  TyT  of  '/  of  !r0.  ; 

i  X   jo  X   to      100     4 
then  --    —      —  =  -  =  -c.  Ans. 
»75X    i    X    i       i?5     7 

2.  Reduce  ^^  of  a  pound  to  the  fraction  of  a  penny. 

Ans    J.J. 

3.  Reduce  5V  of  a  Ib.   Avoirdupois  to  the  fraction  of 
an  ounce.  Ans.  4°z* 

4.  Reduce  r  gV  tf  °f  a  ^'  Troy  to  the  fraction  of  a  pwt. 

Ans. 


5.  Reduce  T^  of  a  cwt.  to  the  fraction  of  a  Ib  Av- 
oirdupois. Ans.  Jib. 


&T  REDUCTION   OF 

CASE    IX. 

To  find  the  value  of  a  fraction  in  the  known  parts   of  the 
iritegery  as  of  colny   weight,  meafure,  &c. 

RULF. 

Multiply  the  numerator  by  the  parts  in  the  next  infe- 
riour  denomination,  and  divide  the  product  by  the  de- 
nominator ;  and  if  any  thing  remain,  multiply  it  by  the 
next  inferiour  denomination,  and  divide  by  the  denomi- 
nator as  before,  and  fo  on,  as  far  as  necefTary  ;  and  the 
quotients  placed  after  one  another,  in  their  order,  will  be 
the  anfwer  required.  Or,  Reduce  the  numerator,  as  if 
it  were  a  whole  number,  to  the  lowed  denomination,  and 
divide  the  refult  by  the  denominator  ;  the  quotient  will 
be  the  number  of  the  lowed  denomination,  (which  mud 
be'brought  into  higher  denominations  as  far  as  it  will  go) 
and  the  remainder  will  be  a  numerator  to  be  placed  over 
the  given  denominator  for  a  fraction  of  the  lowed  denom« 
ination. 

NOTE.  From  this  rule,  in  connection  with  what  has 
been  faid  of  Reduction  of  Federal  Money,  it  appears, 
that,  annexing  to  the  given  numerator  as  many  cyphers, 
as  will  fill  all  the  places  to  the  lowed  denomination,  and  di- 
yidlrig  tnc  number  fo  formed  by  the  denominator,  the  quo^ 
tient  will  be  the  anfwer  in  the  feveral  denominations,  and 
the  remainder  a  numerator  to  be  placed  over  the  given 
denominator,  forming  a  fraction  of  the  lowed  denomi- 
nation. 

EXAMPLES. 

I-.  What  is  the  value  of  |-  of  a  dollar  ? 
By  the  general  rule.  By  the  note, 

5  D.  d.    c.     n>, 

10  8)5    _o_  o  _  o 

8)5o(     2  6   '»     5 

d.  6     10 

Ans.  6d.  2c.  5m: 
8)2o(     4  or  6.2C.  5  my 

C,  2      10 


m. 


VULGAR  FRACTIONS.  65 

Or  thus. 

5D.=5ooom.  and  30-^°m>—62^m.=62c.  51x1.    anfwer,  as 
before. 

2.  What  is  the  value  of  ^  of  a  dollar  ? 

D.    d.    c.  m. 

64)17     o     o     o1  (2d.  6c. 
128 
or  260. 


420 

384  Or,   170=1  yooom.     And 


320 

-  260-.  5^m.  Ans.  as  before, 

40  .< 

-  =  IT 
64 

3.  What  is  the  value  of  ?•  of  a  pound  ? 

Ans   145.  3d. 

4.  What  is  the  value  of  y  of  a  Ib.  Avoirdupois  ? 

Ans.  i2oz.   i2jdr. 

5.  What  is  the  value  of  ^  of  a  mile  ? 

Ans.  6fur.   26pol,  lift. 

6.  What  is  the  value  of  Tpr  of  a  day  ? 

Ans.   i6h.  36111. 

7.  What  is  the  value  of  g-  of  an  acre  ? 

Ans.  3roods 


CASE    X.* 

7"<?  r^<r<?   any  given  quantity  to  the  fraftion  of  any  greater 
denomination  of  the  fame  kind. 

RULE. 

Reduce  the  given  quantity  to  the  loweft  term  mention- 
tioned,  for  a  numer.ttor  ;  then  reduce  the  integral  part 
to  the  fame  term  for  a  denominator  j  which  will  be  the 
fraftion  required. 

*    This  cafe  13  the  reverfe  of  the  former,  therefore  proves  it. 

NOTE.     If  there  !ie  a  fraction    given    with  the   fa:J  qua  itity,  it 
mu:t  be  tarther  reduced  to  the  dcaooiuutive  parts  thereof,  adding 
thereto  the  numerator. 
F  2 


66  REDUCTION  OF 

NOTE.  It  appears  from  this  rule,  and  what  has  been 
faid  before,  that,  in  Federal  Money,  where  the  given 
quantity  contains  no  fraction  of  its  lowed  denomination, 
the  annexing  of  as  many  cyphers  to  i  of  the  required  de- 
nomination, as  will  extend  to  the  loweft  denomination 
in  the  given  quantity,  will  form  a  denominator,  which 
placed  under  the  given  quantity  ufed  as  one  number  for 
a  numerator,  will  make  the  anfwer,  which  may  be  reduc- 
ed to  its  loweft  terms.  Or,  if  there  be  a  fraction  of  the 
loweft  denomination,  multiply  the  given  whole  numbers 
by  its  denominator,  adding  its  numerator,  for  a  numeral- 
or  ;  and  let  the  denominator  itfelf,  at  the  left  of  as  many 
cyphers  as  were  mentioned  above,  be  a  denominator  j 
the  fraction  fo  formed  will  be  the  anfwer  ;  which  may  be 
.reduced  to  its  loweft  terms. 

EXAMPLES. 

i.  Reduce  6d.  2C.  5m.  to  the  fraction  of  a  dollar^ 
By  the  general  rule. 

lod.  int.  part. 
10 

100  By  the  note. 

10  D.  d.     c.     m.. 

-  625 

jooo  --  =|D. 

1000 

Ans.  Ans.  as  before. 


Reduce  *6c.  $^m.  to  the  fraction  of  a  dollar, 

By  the  general  rule. 
26c.  looc.  int.  pt. 

Xio+5 


VULGAR  FRACTIONS.  6; 

By  the  note. 

• D.  d.    c.  m. 

265x84-54-2     i     2     5 


And  iD-x8=8     o     o    o 

Ans.  as  before; 

3.  Reduce  145.  3^d.  4  to  the  fraction  of  a  pound. 

4  8  q  o  _  s  r 


4.  Reduce  1202.   i2^dr.  to  the  fraction  of  alb.  Avoir- 
dupois. Ans.  Jib. 

5.  Reduce  6fur.  26po.   lift,  to  the  fraction  of  a  mile. 

Ans.  |m. 

6.  Reduce  i6h.  36m.  55fVs*  to  the  fraction  of  a  day. 

Ans.  T?day. 

7.  Reduce  3r.   1 7  jpo.  to  the  fraction  of  an  acre. 

Ans.  4  acre.. 


ADDITION  OF   VULGAR  FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  fmgle  ones  ;  mixed 
numbers  to  improper  fractions  ;  fractions  of  different  in- 
tegers to  thofe  of  the  fame  ;  and  all  of  them  to  a  com- 
mon denominator  ;  then,  the  fum  of  the  numerators 
written  over  the  common  denominator  will  be  the  fum  of 
the  fractions  required. 

EXAMPLES. 

i.  Add  7*,  4  of  ¥»  andr7  together. 
Firft,   74=V9,  4  of  4=^-,  and  7=1-. 
Then  the  fractions  are  3T9>  T!»  an^  T  >  therefore,, 
39X5 6x  1=2184 
i5x  5X  i=     75 
7X  5x56=1960 

4219 

=  »5j&fe 

5x56x1=280 

2184-1-754-1960 
Or  thus,  -  


48  SUBTRACTION  OF 

2.  What  is  the  fum  of  TV  of  4!-,  |  of  |-,  and  9!  ? 

Ans.   i2|-f« 

3.  Add  together  ^D.  fc.   T3^c.   and  ^m. 

Ans.  2oc.  9m. 

4.  Add|l  4s-  and  yd.  together.         Ans.  2s   8ryTd. 

5.  Add  ^  of  a  week,  ^  of  a  day,  ^  of  an  hour,  a»d  ^ 
of  a  minute  together. 

Ans.  2  days,  2  hours,  30  minutes,  45  feconds, 

SUBTRACTION  OF  VULGAR  FRACTIONS. 

RULE.* 

Prepare  the  fractions  as  in  Addition,  and  the  difference 
of  the  numerators,  written  above  the  common  denomina- 
tor, will  give  the  difference  of  the  fractions  required. 

EXAMPLES. 
I.  From  |  take  f  of  | 
.*  of  |.=4f =ffs-g..     Then  the  fractions  are  |  and  ~/g. 

S^L4  ll=AV  and  5-r%,  therefore. 


t 


4X28=1  J2  com  den.      J  T8TV  —  rrs^TTB—Z  remainder. 
2.  From  45  take  |.  Ans. 


*  In  fubtra(5ling  mixed  numbers,  when  the  tradlions  have  a  con> 
mon  denominator,  and  the  numerator  in  the  fubtrahend  is  lefs  than 
that  in  the  minuend,  the  difference  pf  the  whole  numbers  will  be  a 
whoie  number,  and  the  difference  of  the  numerators  a  numerator  to 
be  placed  over  the  given  denominator  :  this  whole  number  and  the 
fra<5tion  thus  formed  will  be  the  remainder,  but,  when  the  numera- 
tor in  the  fubtrahend  is  greater  than  that  in  the  minuend,  fubtradl 
the  numerator  in  the  i'ubtrahend  from  the  common  denominator,  ad- 
ding the  numerator  in  the  minuend,  and  carrying  one  to  the  integer 
of  the  fubtrahend. 

Hence,  A  fraftion  is  fubtracted  from  a  whole  number,  by  taking 
the  numerator  of  the  fraction  from  its  denominator,  and  placing 
the  iemainder  over  the  denominator,  then  taking  one  from  the 
whole  number. 

EXAMPLES. 

From   12  J  i2§  12 

Take      7$  7|  § 

Rem.  u 


VULGAR   FRACTIONS, 

3.  Take  3^0.  from  ^of  2^D.  Ans. 

4.  From  £1.  take  r9^s.  Ans.  45 
5".  From  5  weeks  take    19^  days. 

Ans.   i5da.  4ho.  4 


MULTIPLICATION  OF    VULGAR  FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mixed 
numbers  to  improper  fractions.}  then  the  produft  of  the 
numerators  will  be  the  numerator,  and  the  product  of  the 
denominators,  the  denominator  of  the  produdl  required. 

NOTE.  Where  feveral  fractions  are  to  be  multiplied,  if 
the  numerator  of  one  fraction  be  equal  to  the  denominat- 
or of  another,  their  equal  numerators  and  denominators 
may  be  omitted. 

EXAMPLES. 

1.  What  is   the  continued  product  of  4^,  ^-,   ^-of  -j^ 
and  6. 

1X7 

4l=V3aof^—  =TV  and  6=4- 
4X8 

I3XIX  7X6 

Then  yxiX7'T*T=  -----  .ssJ^JssiU  the  anfwer. 
3X5X32X1 

2.  Multiply  T\  by  -^j-.  Ans.  ^TV 

3.  Multiply  5^;  by  £.  Ans.  £. 

4.  Multiply  ^  of  5  by  |  of  -J.  Ans.  TV 


DIVISION  OF  VULGAR  FRACTIONS. 

RULE.* 

Prepare  the  fra&ions  as  before  ;  then,  invert  the  divi- 
for  and  proceed  exadtly  as  in  Multiplication  :  The  pro- 
duds  will  be  the  quotient  required. 

*  To  multiply  a  fra&ion  by  an  integer,  divide  the  denominatofj 
or  multiply  the  numerator  by  it  ;  and  to  divide  by  an  integer,  di- 
vi.de  the  numerator,  or  multiply  the  denominator  by  it-,; 


70  DECIMAL  FRACTIONS. 

EXAMPLES. 

1.  Divide  |  of  17  by  §  of  £. 

1X17 
.Jof  1 7=4  of  V= =y,  and  |  of  i«if=J  ;  there- 

3><i 
17X2 

fore,  y-~i= =  y=n^the  quotient  required. 

3Xi 

2.  Divide  |  by  |.  Ans    i/T. 

3.  Divide  5 1  by  7|.  Ans.  ££. 

4.  Divide  4  by  9.  An?.  ^'T. 
3.  Divide  £  of  £  of  f  by  |  of  £  Ans.  f . 


DECIMAL    FRACTIONS. 

DECIMAL  Fraftions  are  of  fuch  a  nature,  that  they 
vary  in  the  fame  proportion,  and  are  managed  by  the 
fame  method  of  operation,  as  whole  numbers  are. 

On  this  account,  every  proper  fraftion  is  fuppofed  to 
be  reducible  to  another,  whofe  denominator  (hall  be  10, 
TOO,  1000,  &c.  viz.  Unity,  with  a  number  of  cyphers 
annexed  ;  and  fractions  with  fuch  denominators  are  called 
Decimal  Fractions  .  Such  are  f^,  fVV  T6VVV»  &c. 

As  the  denominator  of  a  decimal  fradion  is  always  ic, 
300,  IOQO,  &c.  the  denominators  need  not  be  expreffed  : 
For  the  numerator  only  may  be  mude  to  exprefs  the  true 
va-ue  :  For  this  purpofe  it  is  only  required  to  write  the 
numerator  with  a  point  before  it  at  the  left  hand,  to  dif- 
tin^uifli  it  from  a  whole  number,  when  it  coniifts  of  fo 
m.mv  figures,  as  the  denominator  hath  cyphers  annexed 
to  unity,  or  i  :  So  yV  -«  written  -5  ;  TV<7,  -33  ;  TW^> 
•735,  &c. 

NOTE.     The  point  prefixed  is  called  a  Separatrix. 

But  if  the  numerator  has  not  fo  many  places  as  the 
denominator  has  cyphers,  put  fo  many  cyphers  before  it, 
viz.  at  the  left  hand  as  will  make  up  the  defect  :  So  write 
T^  thus.  -05  ;  and  Tffro  thus,  *oo6,  &c-  And  thus  do 
thefe  fractions  receive  the  form  of  whole  numbers. 


DECIMAL  FRACTIONS.  71 

The  ift,  2d,  3d,  4th,  &c  places  of  decimals,  counting 
from  the  left  hand  toward  the  rigjit,  are  called  primes, 
feconds,  thirds,  fourths,  &c. 

We  may  confider  unity  as  a  fixed  point,  from  whence 
whole  numbers  proceed  infinitely  increafmg  toward  the 
left  hand,  and  decimals  infinitely  decreafmg  toward  the 
right  hand  to  o,  as  in  the  following 

TABLE.* 


«—  '  S3    C    o    °  f^    £  £  »rt    O    o    C  "^  '  —  • 

§l3£fi%|  ajBsiieeJsi 

&&zlr<r*  o  g  j-  -g  «  g  o  H  c-  :=;  £i  2 

UXSoXhEH^HE^XO^XU 
987654321-23456789 


*  It  will  be  very  ?pparent  to  the  learner  from  the  nature  of  deci- 
mals, and  what  has  been  faid  of  Federal  Money,  that  thh  money  is 
purely  decimal  ;  and  the  dollar  being  the  money  unit,  the  lower  de- 
nominations are  plainly  fo  many  decimal  parts  of  a  dollar  ;  thus  9 
dollars  and  8  dimes  are  exprefled  9  8=9_8&  doll—  i  a  dollars,  4  dimes 
and  7  cents  thus,  ia'47  =  ia_^TJ.  dell  —  zo  dollars,  3  dimes,  4  cents, 
and  5  mills,  thus,  20-345=20^^  doll  —  100  dollars  and  9  mills, 
thus,  iooiOO9=iooT._^JTy  doll.—  50  dollars  and  j  cents,  thus,  50-05 
—jOy-Jtf  doH*  ;  whereforc>  i'  is»  »n  all  refpe&s,  added,  fubtra<Sled, 
multiplied,  and  divided,  the  fame  as  decimals  ;  and,  of  all  coins*,  it  is 
the  moft  fimple. 

It  may  atfo  be  obfcrved,  that  the  fum  exhibits  the  particular  num- 
ber of  each  different  piece  of  money  contained  in  it,  viz.  455997 

ents=4559rVtf  dime&=455T9^V  dollars= 
E    D.  d.  c  m. 
=4  5  J-  9  9  7- 


Alfo,  the  names  of  the  coins,  lefs  than  a  dollar,  are  fignificant  of 
their  values.  For  the  mill,  which  (lands  in  the  third  place  at  the  right 
hand  of  the  feparatnx  or  place  of  thoufandths,  rs  contracted  from 
»»//<•,  the  Latin  for  thoufand  :  Cent,  which  occupies  the  fecoud  place, 
or  place  of  hundrcdths,  is  an  abbreviation  of  centum,  the  Latin 
for  hundred  :  And  dim?,  which  is  ia  the  firft  place,  U  deriTed  from 
e,  the  French  for  tentlv. 


•$s  DECIMAL  FRACTIONS. 

From  this  table  it  is  evident,  that,  in  decimals,  as  well 
as  in  whole  numbers,  each  figure  takes  its  value  by  its 
diftance  from  unit's  place  :  If  it  be  in  the  firft  place  af- 
ter units  (or  the  feparating  point)  it  fignifies  tenths ;  if  in 
the  fecond,  hundreds,  &c.  decreafmg  in  each  place  in  a 
tenfold  proportion. 

Confequently,  every  fmgle  figure  expreffing  a  decimal 
has  for  its  denominator  an  unit,  or  i,  with  fo  many  cy- 
phers as  its  place  is  diftant  from  unit's  place  :  Thus  2  in 
the  decimal  part  of  the  table=T2<j  ;  3=T|^  ;  4— nnre>  &c« 
And  if  a  decimal  be  expreiTed  by  feveral  figures,  the  de- 
nominator is  i,  with  fo  many  cyphers  as  the  loweft  figure 
is  diftant  from  unit's  place.  So  -357  fignifies  TVcrV>  an<^ 
'•0053=^^,  &c. 

Cyphersj  placed  at  the  right  hand  of  a  decimal  frac- 
tion, do  not  alter  its  value,  since  every  significant  figure 
continues  to  poffefs  the  fame  place  :  So  -5,  -50,  and  -500^ 
are  all  of  the  fame  value,  and  each  equal  to  \, 

But  cyphers  placed  at  the  left  hand  of  a  decimal,  do 
alter  its  value,  every  cypher  depreffing  it  to  T\j-  of  the 
value  it  had  before,  by  removing  every  fignificant  figure 
one  place  further  from  the  place  ©f  units.  So  -5,  '055 
•005,  all  exprefs  different  decimals,  viz.  '5,  -f^  ;  05,  -j^ ; 

'005,  Tc/W 

Hence  may  be  obferved  the  contrary  effecls  of  cyphers 
being  annexed  to  whole  numbers,  and  decimals. 

It  is  likewife  evident  from  the  table,  that  fmce  the  pla- 
ces of  decimals  decreafe  in  a  tenfold  proportion  from  u- 
nits  downwards,  fo  they  confequently  increafe  in  a  tenfold 
proportion  from  the  right  h*nd  toward  the  left,  as  the 
places  of  whole  numbers  do  :  For,  ten  hundredth  parts 
make  one  tenth,  ten  tenths  makt  i  ;  ten  units,  ten  ;  ten 
tens,  one  hundred,  &c.  viz  -viSr— T\F>  T§='"  anc*  1X10= 
10,  which  proves  that  decimals  are  fubjecT:  to  the  fame 
law  of  Notation,  and  confequently  of  operation,  as  whole 
numbers  are. 

Decimal  fractions  of  unequal  denominators  are  reduc- 
ed to  one  common  denominator,  when  there  are  annexed 

Such  being  the  nature  of  Federal  Money,  its  operations  can  in  no 
other  way  be  fo  well  underftood,  as  in  obtaining  a  good  knowledge 
of  Decimals.  <uid  applying  their  feveral  rules  to  the  various  cafe?  of 
money  matters. 


ADDITION  OF  DECIMALS.  73 

to  the  right  hand  of  thofc,  which  have  fewer  places,  ib 
many  cyphers  as  make  them  equal  in  places  with  that 
which  has  the  moft.  So  thele  decimals,  '5,  *c6,  '455> 
may  be  reduced  to  the  decimals  '500,  *o6o,  '455,  which 
have  all  1000  for  their  denominator 

Of  decimals,  that  is  the  greateft,  whofe  higheft  figure 
is  greateft,  whether  they  confift  of  an  equal  or  unequal 
number  of  places  :  Thus,  '5  is  greater  than  '459, 
for  if  it  be  reduced  to  the  fame  denominator  with  -459,  it 
will  be  -500. 

A  mixed  immber,  viz.  a  whole  number  with  a  decimal 
annexed,  is  equal  to  an  improper  fraction,  whofe  numera- 
tor is  all  the  figures  cf  the  mixed  number,  taken  as  one 
whole  number,  and  the  denominator  that  of  the  decimal 
part  So  45  309  is  equal  to  VoVV.  as  *s  evident  from 
the  method  given  to  reduce  a  mixed  number  to  an  im- 
proper fraction  :  Thus,  45X1 000^-3 09= V^W  as  above, 


ADDITION  OF  DECIMALS. 

RULE. 

1.  Place   the  numbers,  whether  mixed,  or  pure  decim- 
als, under  each  other,   according  to  the  value  of  their 
places. 

2.  Find  their  fum,  as  in  whole  numbers,  and  point  off 
fo  many  places  for  decimals,  as  are  equal  to  the  greateft 
number  of  decimal  places  in  any  of  the  given  numbers. 

EXAMPLES. 

i.  Find   the  fum  of  19  o73+ 
3478-1+12  358.- 

19073 

2  3597 
223 

-o.  97581 
3+7*  ' 

1235S 

3734<;iC4c8i   the  fum. 


74  SUBTRACTION,  MULTIPLICATION  AND 

2.  Required   the  fum   of  429+2 1*37+35 S'oc^- 

1-7  ?  Ans,  808- 143. 

3.  Required    the    fum     of     973+ 19-**  7 5+93 '7164-'- 
•9501  ?  Ans.   1088-4165. 

SUBTRACTION  OF  DECIMALS. 

RULE. 

Place  the  numbers  according  to  their  value  :  Then 
iubtract  as  in  whole  numbers,  and  point  off  the  decimals 
as  in  Addition. 

EXAMPLES. 

i.  Find  the  difference  of  1793*13  and  817*05693. 
From   1 793- 13 
Take      817-05693 


Remainder  976-07307 

2.  From    1 71- 1 95  take  125  9176.  Ans.  45-2774. 

3.  From  219-1384  take  195*91.  Ans.   23-2284. 

4.  From  480  take  245*0075.  '  Ans.  234-9925. 

MULTIPLICATION  OF  DECIMALS. 
RULE 

1.  Whether  they  be  mixed  numbers,  or  pure  decimals, 
place  the  factors  and  multiply  them  as  in  whole  numbers. 

2.  Point  off  fo  many  figures  from  the  product  as  there 
are  decimal  places  in  both  the  factors  ;  and  if  there  be 
not  fo  many  places  in  the  product,  fupply  the  defect  by 
prefixing  cyphers. 

EXAMPLES. 
i.  Multiply    '02345 
by  -00163 

7035 
14070 

2345 


•0000382235  the  product. 


2-.  Multiply  25-238  by  12-17.  Ans.  307*14646. 

3.  Multiply  -3759  by  -945,  Ans.  -355 22 55. 

4,  Multiply  '84179  by  -0385.  Ans.  -032408915. 


DIVISION  OF  DECIMALS.  75 

To  multiply  by  10,  100,  1000,  &c.  remove  the  feparat- 
jug  point  fo  many  places  to  the  right  hand,  as  the  multi- 
plier has  cyphers. 

i"'o      )  f345 

80  '345  multiplied  by  <  100    >  makes  <  34-5 

tioooj  (.345 


For  '345x10  is  3'45Q>  &c- 


DIVISION  OF  DECIMALS. 

RULE. 

r.  The  places  of  decimal  parts  in  the  divifor  and  quo- 
tient, counted  together,  muft  always  be  equal  to  th.Ms  in 
the  dividend  ;  therefore,  divide  as  ia  whole  numbers,  and, 
from  the  right  hand  of  the  quotient  point  off  fo  many- 
places  for  decimals,  as  the  decimal  places  in  the  dividend 
exceed  thofe  in  the  divifor. 

2.  If  the  places  of  the  quotient  be  not  fo  many  as  the 
rule  requires,  fupply  the  defect  by  prefixing  cyphers  to 
the  left  hand.  * 

3  If  at  any  time  there  be  a  remainder,  or  the  decimal 
places  in  the  di  vrifor  be  more  than  thofe  in  the  dividend, 
cyphers  may  be  annexed  to  the  dividend  or  to  the  re- 
mainder, and  the  quotient  carried  on  to  any  degree  oir 
£Xa<flnefs. 

EXAMPLES. 


23. 


76  DIVISION  OF   DECIMALS. 

2. 


oi'iyS,  &c, 
37r9 

8103 
7438 

6620 
37/9 

29010 
26033 

29770 
29752 

18 

In  Example  id,  the  divifor  having  no  decimals,  tke; 
quotient  muft  have  fo  many  as  there  are  in  the  dividend. 
In  Exfmple  2d,  the  dividend  being  an  integer,  muft  have 
at  leaft  fo  many  cyphers  annexed.  as  there  are  decimal  5 
HI  the  divifor,  and  fo  far  the  quotient  will  be  whole  num. 
bers  ;  then  annexing  more  cyphers,  the  remaining  figures 
in  the  quotient  will  be  decimals,  according  to  tfre  rule. 

3- 

4- 
5. 


?•    : 

When  decimals  or  whole  numbers  are  to  be  divided  by 
10,  100,  i coo,  fee.  (viz.  unity  with  cyphers)  it  is  per- 
formed by  removing  the  feparatrix,  in  the  dividend,  fo 
many  places  toward  the  left  hand,  as  there  are  cyphers  ia 
the  divifor. 


10? 

100  I    ,.  . 
ooofdlv" 

»COOJ 


So   <    1(^  ^  dividing  7654,  the  quotient  is 

LI oooo 


*  The  foi'owing  queftions  arc  left  unpointed  in  the  quotient,  to 
cxcrcife  the  learner. 


REDUCTION  OF  DECIMALS.  7; 

REDUCTION  OF  DECIMALS. 

CASE  I. 

To  reduce  a  vufgar  fraflion  to  its  equivalent  decimal. 

RULE. 

Divide  the  numerator  by  the  denominator,  as  in  Divii- 
ion  of  Decimals,  and  the  quotient  will  be  the  decimal  re- 
quired— Or,  So  many  cyphers  as  you  annex  to  the  given 
numerator,  fo  many  places  muft  be  pointed  off  in  the 
quotient,  and  if  there  be  not  fo  many  places  of  figures 
in  the  quotient,  the  deficiency  mull  be  fupplied  by  pre- 
fixing fo  many  cyphers  before  the  quotient  figures. 

EXAMPLES, 

1.  Reduce  \  to  a  decimal.  S)rooo 

•125  Ans. 

2.  Reduce  -J,  -J,  and  y  to  decimals. 

Ans.  -375,  -625,  -666-K 

3.  Reduce  £,  j>  |,  ^,  f,  -J-»  and  -g-  to  decimals. 

Ans.  -25,  -5,  -75,  -333+,  -8,  -833+,  -875. 

4.  Reduce  T5¥,  fj,  ^y,  and  -fa  to  decimals. 

Ans.  -2634-,  -692-^,  -025,  -25, 

CASE  II. 

To  reduce  numbers   of  different  denominations^   as  of  Money, 
Weighty  and  Meafurey  to  their  equivalent  decimal  values*      * 

RULE. 

1.  Write  the  given  numbers  perpendicularly  under  each 
other,  for  dividends,  proceeding  orderly  from  the  leatt  to 
the  greateft. 

2.  Oppofite  to  each  dividend,  on  the  left  hand,   place 
fuch  a  number,  for  a  divifor,  as  will  bring   it  to  the  next 
fuperiour  denomination,  and  draw  a  line  perpendicularly 
between  them. 

3.  Begin  with  the  higheft,  and  write  the  quotient  of 
each  divifion  as  decimal  parts  on  the  right  hand   of  the 
dividend  next  below  it,  and  fo  on,  until  they  are  all  ufed, 
and  the  laft  quotient  will  be  the  decimal  fought, 

G  2 


REDUCTION  OF  DECIMALS. 

EXAMPLES. 

r.  Reduce  175.  8|d.  to  the  decimal  of  a  pound. 
4     3* 


•886458,  &c.  the  decimal  required. 

Here,  in  dividing  3  by  4,  I  fuppofe  two  cyphers  to  be 
annexed  to  the  3,  which  make  it  S'oo,  and  "75  is  the  quo- 
tient, which  I  write  againft  8  in  the  next  line  ;  this  quo- 
tient, viz.  8-75,  being  pence  and  decimal  parts  of  a  pen- 
ny, I  divide  them  by  12,  which  brings  them  to  (hillings 
and  decimal  parts  ;  I  therefore  divide  by  20.  and  (there 
being  no  whole  number)  the  quotient  is  decimal  parts  of 
a  pound. 

2.  Reduce  i,  2,  3,  4,  and  fo  on  to  19  (hillings,  to 
decimals. 

Shillings.          i  2  3456 

Answers.      -05,          -i,         '15,         -2        -25,        -3, 

Shillings.         7          8  9  10  n  12 

Anfwers.      '35,       '4,       '45,  '5,         -55,  -6, 

Shillings.  1 3  14  15  -16  17  18  19 
Ans.  -65,  -7,  -75,  -8,  -85,  -9,  -95. 
Here,  when  the  (Hillings  are  even,  half  the  number, 
with  a  point  prefixed,  is  their  decimal  expreffion  ;  but  if 
the  number  be  odd,  annex  a  cypher  to  the  (hillings,  rind 
then  halving  them,  you  will  have  their  decimal  exprefllon* 

g.  *Reduce  i,  2,  3,  and  fo  on  to  1 1  pence,  to  the  dec- 
imals of  a  (hilling. 

Pence.         123456 
Anfwers.   '083+,  -166,        -25,       '333+,     '416+,      -5, 

Pence.         7  8  9  10  n 

Anfwers.  '583+,       '666+,  '75,  '833+,      '9164-. 

4.  Reduce  i,  2,  3,  &c.  to  1 1  pence,  to  the  decimals  of 
a  pound. 

*  The  anfwers  to  this  qucTUon  are  the  fame  as  the  decimal  p^rts 
of  a  foot. 


REDUCTION  OF  DECIMALS.  7$ 


Pence.         i  2  34 

Anfwers.  -00416+,         '00834-,          '0125,         -01666+. 

Pence.          567  8 

Anfwers.     '0208+,         '025,         '029164-,          '03334-, 

Pence.         9  10.  1 1. 

Anfwers.     "0375,  "04164-,  -045834-. 

5.  Reduce  i,  2,  and   3  farthings  to  the   decimals  of  a 
penny.  Ans.  iqr.='25d.   2qrs.='5d.  and  3qrs.='75d. 

6.  Reduce  i,  2,  and  3  farthings   to  the  decimals  of  a 
(hilling. 

Ans.   i qr.=s-  020834-5    2qrs. ='041664-5.   3q~s  ='06255. 

7.  Reduce  ;,  2,  and   3    farthings  to  the  decimals  of  a 
pound. 

Ans.    I  qr.srjf '00104164-      2qrs. =£.0020834-    3qrs.=: 
£-003125. 

8.  Reduce  133.  jjd.  to  the  decimal  of  a  pound. 

Ans.  -67294-. 

9.  Reduce   7cwt.    3qrs.  I7lb.  looz.  I2dr.    to  the  deci- 
mal of  a  ton.  Ans.   -395384-. 

10.  Reduce    icoz.    I3pwt.    9gr.  to    the  decimal  of  a 
pound  Troy.  Ans.  '8890625. 

11.  Reduce  5qrs.  311.  to  the  decimal  of  a  yard. 

Ans.  -9375. 
CASE    III. 

To  Jind  the  decimal  of  any  number    of  {killing /,  pence >   and 
far  things  i  by  infpeclion. 
RULE. 

1 .  Write  half  the  greateft  even  number  of  /hillings  for 
the  firft  decimal  figure. 

2.  Let  the  farthings  in   the   given  pence  and  farthings 
poflefs  the  fecond  and  third  places  ;  obferving  to  increafe 
the  iecond  place,  or  place  of  hundredths,  by  5,  if  the  fhil- 
lings  be  odd,  and  the  third  place  by  i,  when  the  farthings 
exceed  12,  and  by  2  when  they  exceed  36. 

EXAMPLES. 
i.  Find  the  decimal  of  135.  9|d.  by  infpeclion. 

•6  .  .    =J  Of    I2S. 

5  for  the  odd  (hilling. 
39  =  farthings  in  9^d. 
Add      2  for  the  excefs  of  36. 

•691  —  decimal  required. 


Be  REDUCTION  OF  DECIMALS; 

2,  Find  by  infpection  the  decimal  expreffions  of 
3^d.  and  iys.  8^d.  Ans. 


CASE  IV. 

To  fnd  tfje  value  of  any  given  decimal  in  the  terms  cf  the 
integer. 

RULE.— i.  Multiply  the  decimal  by  the  number  of 
parts  in  the  next  lefs  denomination,  and  cut  off  fo  many 
places  for  a  remainder,  to  the  right  hand,  as  there  are 
places  in  the  given  decimal. 

2.  Multiply  the  remainder  by  the  next   inferiour  de- 
nomination, and  cut  off  a  remainder  as  before. 

3.  Proceed  in  this  manner  through  all  the  parts  of  the 
integer,  and  the  feveral  denominations,   (landing   on  the 
left  hand,  make  the  anfwer. 

EXAMPLES. 

i.  Find  the  value  of  73968  of  a  pound. 
20 


2*09280        Ans.  145.  9^d. 

2.  What  is  the  value  of  '679  of  a  (hilling  ? 

A  -fir-          O  •  T     •  C    -* 

xHib»    o  J AS Gv 

3.  What  is  the  value  of  -617  of  acwt.  ? 

Ans.  2qrs.  I3lb.  102.  io-^dr. 

4.  What  is  the  value  of  -397  of  a  yard  ? 

Ans.    iqr.  2-352^. 

7.  What  is  the  value  of  '8469  of  a  degree  ? 

Ans.  58m.  6fur.  35po.  oft.  nin. 

8.  What  is  the  value  of  '569  of  a  year  ? 

Ans.  207<i,  i6h.  26m,  24fejr, 


REDUCTION  OF  DECIMALS.  8t 


CASE    V. 
2"*  find  the  value  of  any  decimal  of  a  pound  by 

RULE. 

Double  the  firft  figure,  or  place  of  tenths,  for  fhillings, 
and  if  the  fecond  figure  be  5,  or  more  than  5,  reckon  an- 
other fhiUing  ;  then,  after  the  5  is  deducted,  call  the  fig- 
ures  in  the  fecond  and  third  places  fo  many  farthings, 
abating  i  when  they  are  above  12,  and  2  when  above  36, 
andthj  refult  will  be  the  anfwer. 

NOTE  When  the  decimal  has  but  two  figures,  if  any 
thing  remain  after  the  Shillings  are  taken  out,  a  cypher 
muft  be  annexed  to  the  right  hand,  or  fuppofed  to  be  fo. 

EXAMPLES. 

1.  Find  the  value  of  £876  by  infpection. 

1 6s. = double  of  8 
is.  for  the  5  in  the  fecond  place,   which  is 

to  be  taken  out  of  7, 

and  6jd.=z6  farthings,  remain  to  be  added  ; 

deduct  id.  for  the  excefs  of  12. 

17*.  6£d.  Ans. 

±.  Find  by  infpection,  the  value  of  j£*49' 
bs.=double  of  4. 
is.  for  the  5  in  the  place  of  hundredths. 

i  cd.  =40  farthings,  a  o  being  annexed  to   the  re- 
maining 4. 
Dcd.         id.  for  the  excefs  of  36. 

95.  9id.  the  anfwer. 

2.  Find  the  value  of  £'097  by  infpecYion. 

Ans.  is.   u£d. 

4.  Value   the   following  decimals,   by  tnfpection,  and 
find   their   fum,    viz.   £•  7 85+^-5 3 7+^-9 1 6+£-j $+£ -5+ 
-o8.  Ans.  £     i6s,  6d. 


DECIMAL  TABLES. 


DECIMAL  TABLES  of  Coin,  Weight,  and 

Meafure. 
Decimals. 
•010416 

•008333 
•00625 
•004166 
•002083 

TABLE  I.     COIN.  Farthings 
£i  the  Integer.               3 

.1  Decimals 
•0625 
•041666 

I  -020-833 

Grains 
5 
4 
3 

Shil.  :dec.  Shil.  !  dec. 

*9   j   '95i     9      '45 
18      .9        8      -4 
17    !  '85      7      -35 
16      -8  |     6     -3 
ij      '75      5      '25 
14       7         4      -2 
13      '65      3      T5 
12      -6        a     •! 
ii      -55      i      -05 
10      -5 

•* 
i 

TABLE  ill. 
TROY   WEIGHT. 
I  Pound  the  Integer 
Ounces  the  fame 
as  Tab.  II. 

2 

TABLE  IV. 
AVOIRDUPOIS  WT. 
i  i2lb.  the  Integer. 

Penny- 
weights. 

i 

6 
5 
4 
3 
a 
i 

Decimals 

•041666 

•0375 
•0333.13 
•029166 
•025 
•020833 
•016666 
•0125 

•008333 

•004166 

Qrs. 

3 

2 

I 

Poands 

27 

26 

25 

24 
23 

22 
21 

Decimals. 
•75 
•5 

•25 

Pence. 
ii 

TO 

9 
8 

6 
5 

4 
3 
a 

i 

Decimals. 
•045833 
•041666 
•0375 
•033333 
•029166 
•025 
•020833 
•016666 
•0125 
•008333 
•004166 

Decimals. 

•241071 
•23:4143 
•223214 
•214286 
•205357 
•196428 
•1875 
•178571 
•169643 
•160714 
•151786 
•142857 
T33928 
•125 
•116071 
•107143 
•098214 
•089286 
•080357 
•071428 
•0625 
•05357  i 
•044643 
•0357M 
•026786 
•017857 
•008928 

Grains. 
12 
II 
10 

7 
6 
5 
4 
3 
a 
i 

Decimal 
•002083 
•00191 
•001736 
•001562 
•001389 
•001215 
•001042 
•000868 
•000694 
•000521 
•000347 
•000173 

2O 
'9 

18 

17 
16 

15 
14 

13 

12 
1  1 
10 

I 

7 
6 

5 
4 
3 
2 

I 

Farth. 

3 
a 
r 

Decimals. 
•003125 
•0020833 
•0010416 

TABLE  II. 
COIN  &  LONG  MEAS 
I  Shilling  £  i  Foot 
the  Integer. 

Pence  & 
Inches. 
ii 

10 

9 
8 

7 
6 

5 
4 
3 
a 
i 

Decimals. 

•916666 
•833333 
75 
•666666 

•583333 

•416666 
•333333 
•25 
•166666 
•083333 

i  Oz.  the  Integer. 
Pennyweights 
he  fame  as  Shillings 
in  Tab.  I. 

Grains, 
i  A 
ii 

10 

9 
8 

6 

Decimals 

•025 
•022916 
•020833 
•01875 
•016666 

•014583 

•0125 

Ounces 
15 
14 
13 
ia 

Carrie 

Decimals. 
•008370 
•007812 

•007254 
•006696 
I  over. 

DECIMAL  TABLES. 


Brought  ov;-r 

Drams. 

6 
5 
4 

3 

a 

Decimals. 
•©27344 

•023437 
•019531 
•015625 
•011718 
•007812 
•00.7906 

Yards. 
7 
6 

5 
4 
3 

2 

Decimals. 
•003977 
•003409 
•00284  i 
•002273 
•001705 
•001136 
•000568 

Ounces, 
ii 

10 

I 

6 

5 
4 
3 

a 

x 

Decimals. 
•006138 

•00558 
•005022 
•004464 
•003906 
•003348 
•00279 
•002232 
•001674 
•ooi  116 
•000558 

TABLE  VI. 

CLOTH  MEASURE. 
i   Yard  the   Integer. 

Feet 

2 

I 

Inches. 
6 
5 
4 

3 

2 

I 

Decimal-. 

•0003787 
•000189. 
Decimals 
•0000947 
•000079 
•000063: 
•000047* 
•00003  »  6 
•ooooi5v' 

Quarters. 

3 
a 

Decimals. 
75 
'5 
•*5 

qr  s.of  ozs. 
3    . 

i 

Decimals. 
•000418 
•000279 
•000139 

Nails. 
3 

2 

I 

Decimals. 
•1875 
•125 

•0625 

TABLE  V. 
AVOIRDUPOIS  WT. 
lib.  the  Integer. 

TAB 
LIQUID 

r  Gall 
Quarts 
qrs.  ii 
I  Pint 
3  Gill 

2 

I 

LE  VIII 
MEASURE 
the  Integer. 
the  fame  as 
i  Tab.  VI. 

•"5  
•09375 
•0625 
•03125 

TABLE  VII. 
LONG  MEASURE. 
T   Mile  the  Integer. 

Ounces. 
15 
14 
U 
12 
II 
10 

9 
8 

7 
6 

5 
4 
3 
2 

I 

Decimals. 
'9375 
•875 
•8125 
•75 
•6875 
•625 
•5625 
'5 
'4375 
•375 
•31*5 
•*5 
•1875 
•125 
•0625 

Yards. 
IOOO 

900 

800 
700 

600 

500 
400 
300 

200 
100 

90 
80 
70 
60 
50 
40 
30 

20 
IO 

9 

Decimals. 
•568182 
•5"364 
'454545 
'3977*7 
•340909 

•284091 

•2*7272 

•170455 
•113636 
•056818 
•051136 
•045454 
•039773 
•034091 
•028409 
•022727 
•017045 
•011364 
•005682 
•005114 
004549 

TABLE  IX. 
TIME. 
I  Year  the  Integer. 
Months  the  fame  as 
Pence  in  Tab.  II 

Days 

365 
300 
200 
JOO 
90 
80 
70 
60 

50 
40 

L  Carri 

Decimals 

I'OOOOOn 

•82r9iT: 
'547945 
•*7397.-» 
•246575 
•219178 
•191781 
•164384 
•136986  . 
•109589 
ed  over.       I 

Drams.  ' 
15 
14 
13 

12 

II 

IO 

9 

8 

Decimals. 
•058594 
•054687 
•050781 
•046875 
•042969 
•039062 
•035156 
oyt*S 

S4         COMPOUND  MULTIPLICATION. 


Brought  over.        |  Hours. 

Decimals. 

Hours. 

Decimals. 

D*ys.    Decimals. 

21 

•875 

3 

•J*5 

30        -082192 
20'  .  -054795 

2O 
19 

•833333 
•791666 

2 

I 

•083333 
•041666 

10 

9 
8 

7 
6 

5 
4 
3 
•I 
i 

•o*7397 
•024658 
•021918 
•019178 
•016438 
•013699 
•010959 
•008419 
•005479 

•OO27A 

18 

17 
16 

15 
14 
13 
12 
II 
IO 

9 

•75 
•708333 
•666666 
•625 
•583333 
•541666 

*5 
•458333 
•416666 

775 

Minutes. 
30 

20 
£0 

9 
8 

6 
5 

Decimals. 
•020833 
•013889 
•006944 
-00625 
•005555 
•004861 
•004166 
•003472 

i  Day  the  Integer. 

8 
7 
6 

•333333 
•291666 

•2< 

4 
3 
a 

•002777 
•0(^2083 
•001389 

Hours. 

Decimals. 

23 

•958333 

5     !  -208333 

i 

•000694 

22 

•9.6666 

4        -166666 

COMPOUND  MULTIPLICATION 

IS  extremely  ufeful  in  finding  the  value  of  goods,  &c. 
And,  as  in  Compound  Addition,  we  carry  from  the  ;ow- 
eft  denomination  to  the  next  higher,  fo  we  begin  and  car- 
ry  in  Compound  Multiplication  ;  one  general  rule  being 
to  multiply  the  price  by  the  quantity. 

CASE  I. 

Whsn  the  quantity  does  not  exceed  \  2  yards,  pounds  £sV. — 
Set  down  the  price  of  i,  and  place  the  quantity  under- 
neath the  leaft  denomination,  for  the  multiplier,  and  in 
multiplying  by  it,  obftrve  the  fame  rules  for  carrying  from 
one  denomination  to  another,  as  in  Compound  Addition. 

INTRODUCTORY  EXAMPLES. 
i.  2.  3. 

£      s.     d.  £      s     d.          D.   d.    c  m, 

Multiply  15    17     i  25    12    8  8517 

by  2  3  4 


COMPOUND  MULTIPLICATION.        85 

4.  5.  6. 

£.      s.     d.  D.    c.  £.      s.      d. 


67     18     6*  4     75  13     12     ii 

5  6  7 


8. 
E.  D.    d.   c.    m. 

27891 

9 


PRACTICAL   QUESTIONS. 

NOTE  The  facility  of  reckoning  in  the  Federal  Mo. 
jaey,  compared  with  pounds,  (hillings.,  &c  may  be  feen 
from  the  examples  in  this  and  the  following  cafes  ;  where 
the  tame  queitions  are  given  in  both  the  currencies,  as 
near  as  can  be,  avoiding  fmall  fractions.  It  may  be  ob- 
ferved,  that  the  variety  of  cafes  here  given,  is  applicable 
only  to  the  old  currency,  while  the  fame  queftions  in  the 
"Federal  are  folved  by  plain  decimals. 

i.  What  will  9  yards  of  cloth  at    |8|*'  4m*  j  per  yd. 

come  to  ? 

£  o  5  4  price  of  one  yard,         -88c.  9m. 
Multiplied  by  9  yards  9 

Ans.   £,2  8  o  price  of  9  yards.     D-8  -oo     i 


r                       C     9S      I0d           I  _\£*       198. 

(  D.i  630  9m.  3  "  1  D  9  83c  4m. 

4.  9     -     -      ['3*.   7*L       I  .    .    =S£6     *     74 

(D.2  27c.'m  j  tU.20  -43  9 


CASE  II. 

When  the  multiplier  >  that  is.  the  quantity,  is  above  12.- 
You  mult  multiply  by  two  fuch  numbers,  as,  when  multi- 
plied together,  will  produce  the  given  quantity. 


COMPOUND    MULTIPLICATION. 


EXAMPLES. 

i.  What  will  1 44yds.  of  cloth  coft  at  i  3s-      5?d- 

l57c*    6ym 

per  yard  t 

£     s.     d.  c.  m. 

o     3     5-2  price  of  i  yard.  '5764 

Multiply  by                12  144 


Produces          2     i     6  price  of  1 2  yards.  23056 

Multiplied  by  12  23056 

5764 

Anfwer       £24  18     o  price  of  144  yards.     • 

Ans.  D  83-0016 
Queftions.  Anfwers. 


6s.  3|-d. 
D  i  5c.  2m. 
9&-   lod. 
D.I  630.9111 
* 


D.26c. 


per  yd.  =  - 


£l     ii     6 
D  25-24  8 
5     6 


4 
.D.90-75 


CASE     III. 

the  quantity  is  fuch  a  number ^  as  that  no  two  num- 
bers in  the  table  our'//  product  it  exaflfy  :  Then  multiply  by 
two  fuch  numbers  as  come  the  neareft  to  it ;  and  for  the 
number  wanting,  multiply  the  given  price  of  i  yard  by 
the  laid  number  of  yards  wanting,  and  add  the  products 
together  for  the  anfwer  ;  but  if  the  product  of  the  two 
numbers  exceed  the  given  quantity,  then  find  the  value 
of  the  overplus,  which  fubtract  from  the  laft  product,  and 
the  remainder  will  be  the  anfwer. 
EXAMPLES, 

i.  What  will  47  yards  of  cloth,  at    f  175.  9d. 
per  yard,  come  to  ?  ^D.2  95c    8m. 

£o     17     9  price  of  i  yard.  D.2 -95 8 

Multiplied  by  5  47 


Produces        4 
Multiplied  by 

8 

9 

price  of  5  yards. 

price  of  45       Ans. 
price  of  2  yards. 

20706 
11832 

Produces       39 
Add        i 

18 
15 

9 
6 

D.  139-026 

Ans.  £4 1     14    3  price  of  47  yards. 


COMPOUND  MULTIPLICATION.          87 

NOTE.  This  may  be  performed  by  firft  finding  the 
value  of  48  yards,  from  which  if  you  fubtraft  the  price 
of  i,  the  remainder  will  be  the  anfwer  as  above. 

Queftions.  Anfwers. 


3,  6?i j'6s.  3td-  4      |    _      Jf54    l8    34 

-0    J     9'-     7d.T      7  {£28 

ID.I  59c.  7m.  I  ID  94' 


5     5 
223 


CASE   IV. 

WA?«  the  quantity  is  any  number  above  the  Multiplication 
Tabk :  Multiply  the  price  of  i  yard  by  10,  which  A  ill 
produce  the  price  of  10  yards  :  This  product,  multiplied 
by  10,  will  give  the  price  of  100  yards  ;  then,  you  muft 
multiply  the  price  of  100  by  the  number  of  hundreds  in 
in  your  queftion  ;  the  price  often  by  the  number  of  ttns  ; 
and  the  price  of  unity,  or  i,  by  the  number  of  units  : 
Laftly,  add  thefe  feveral  produces  together,  and  the  fum 
will  be  the  anfwer. 

*        EXAMPLES. 


I. 

yard, 

What  will  35-9 
amount  to  ? 

yards  of 

£ 

0 

cloth,  at  f  45.  ^ 
\77C. 
s.    d. 
4     7^  price  c 
10 

'£.}  p- 

f  i   yard. 

6     3     price  of  10  yards. 
10 


23     2     6  price  of  ico  yards. 
3 


69     7     6  price  of  300  yards. 

s  times  the  pric-i  of  ioyds.=n  si     3  price  of    50  yards. 
9  times  the  price  of  i  yd    =217^  price  of    9  yards. 

yards, 


88        COMPOUND  MULTIPLICATION; 


Qaeftions. 


Ans.  0.276*789 


Anfwers. 

!  £*S*  '5  7* 
I  0.855-954 


f- 

4.  512 


_    I  9»-    JItd         1       _ 

|D.i65c.6£m,  J 


{155.   lod. 
O.2  63c.  9m.  J 


_     I 
l 


0.783-406$ 

£4Q5    6     8 
D.i35i-i68 


CASE    V. 

To  find  the  value  of  one  hundred  weight :  As  1 1  z  is  the 
grofs  hundred,  fo  112  farthings  are  =25.  4d.  and  ii2d.= 
95.  4d.  ;  therefore,  if  the  price  be  farthings,  or  not  more 
than  3d.  multiply  2s.  4d  by  the  farthings  in  the  price  of 
lib.  ;  or,  if  the  price  be  pence,  multiply  95,  4d.  by  the 
pence  in  the  price  of  lib.  and,  in  either  «afe,  the  product 
will  be  the  anfwer. 


I.  What  will 
per  pound  ? 

1 1 2  farthings=£o 
lid.  = 


EXAMPLES. 
icwt.  of  chalk  come 


to, 


4  price  of  icwt.  at  iqr.  per  Ib. 
6  farthings  in  the  price. 


Anfwer    £o  14    o  price  of  icwt.  at  i£d  per  Ib. 


1  12 


21 
Anfwer  0.2-352 


COMPOUND  MULTIPLICATION.         39 
i.  iCwt.  of  tin,  atj^  ,xm  j    per  pound? 


•03125  2s.  4d.  price  of  icwt.  at  Jd.  per  Ib. 

112  9  farthings  in  the  price  of  lib. 

6250  £i    i  o  price  of  icwt.  at  2~d.  per  pound> 
3125  anfwer. 


D.  3  50000  Ans. 
3.   iCwt.  of  lead,   at  j^J^  ]  per  pound  ? 

•098  95.  4d.  price  of  icwt.  at  id.  per  lb» 

112  7  pence  in  the  price  of  lib. 

196  £3  5  4  price  of  icwt.  at  yd.  per  Ib, 

98 

98 

0.10976  Ans. 

Queftions.  Anfwers. 

at  {Ic.f4m.}per  pounds  {^.7- 


lqr. 


r  4^-   i          .  $£*  2s. 

--      I     6ic.  -  lD.7 


CASE    VI. 

To  find  the  value  of  two,  or  more  hundred^  by  having  the 

{rice  of  one  pound  :     Firft,  find  the  price  of  ic\vt.   f. .    che 
ift  cafe,  and  then  proceed  to  find  ih-  value  of  khe  whole, 
by  Cafe  i  or  2,  as  the  queftion  may  require. 

H  2 


9Q        COMPOUND  MULTIPLICATION. 

EXAMPLES. 

i .  What  is  the  value  of  5^cwt.  of  fugar, at  f     6d. 
per  pound  ?  £8 

£    s    d. 

094  price  of   icwt.  at  id.  per  Ib*. 
6 

216     o  price  of  ditto  at  6d.  per  Ib, 
5 

14     o     o  price  of  $cwt. 
jqr.  =     o  14     o  price  of  ^  cwt, 

Ans.  £14  14    o  price  of  5^cwt. 

cwt.  qr.  D. 

5      i  -083^ 

4  588 

2 1  664 

28  664 


168  Iy6 

42  - 

588  Ib, 

QueftionS/  Anfwers. 

2.  4Cwt.  of  fugar,  at  I  2Y^'   Iperlb.  =  1^",    \ 

f  ?d.     7  I/*IQ    16   8 


2.  b£  "  j     7C.      f |  D.66  6.  .c. 


-  -  I0 


CASE    VII. 

To  find  4k  f  value  of  a  hundred  weight*  when  the  price  of  i 
Ib.  is  any  number  of  pounds  and  Jbii  lings  ;  or  Jbill  ings,  pence, 
and  farthings  ;  Multiply  the  price  ol  lib.  by  7^  its  prod- 
ucl:  by  8,  and  this  produft  by  2  ;  whicii  laft  produd  will 
be  the  anfwer  required. 


COMPOUND  MULTIPLICATION.         9i 

EXAMPLES. 

What  will  icwt.  of   tobacco  coft,  at  ("     55.    j{d,  1 

per  lb.   ?  i  93c.  ^m.  J 
£     s.     d.  D. 

0     5     7i  price  of  lib.  -9375 

7  112 


i   19     4^  price  of  7lb. 


15   15     o  price  of  561b.  or^cwt. 

2  Ans.   D.IOJ* 

£31    10     o  Ans.  price  of  H2lb.  or  icwt. 
Queftions.  Ar.fwers. 


2    i  _  s-  <5d.      7  _   T       *53  4s. 

ID.I  58ic.j  -  ID  i77  33 

.  _      fi6s.  i  i^d.       1  r   £94  19  4 

|D.2  82c.  6m.  J  '    (.0.316-512 

• 

PRACTICAL  QUESTIONS  IN  WEIGHTS  AND  MEASURES? 

i  .  What  is  the  weight  of  4  hogfheads  of  fugar,  each 
weighing  7cwt.  3qrs.  I9lb.  ?          Ans.  3  icwt.  2qrs.  2olb. 

2.  What  is  the  weight  of  6  chelis  of  tea,  each  weigh- 
ing 3cwt.  2qrs.  9lb.  ?  Ans.  2  icwt.  iqr   261b. 

3.  If  I  anrpoiTeflsd  of  i|  dozen  of  filver  fpoons,  each 
weighing  307.  5pwt.  —  2  dozen  of  tea  ipoons,  each  weigh- 
ing   ijpwt.    I4gr.  —  3   filver  cans,    9r>z.    7pwt  —  2  filver 
tankards,  each    2ioz.  I5pwt.  —  and  6    filver   porringers, 
each  iioz.  i8pwt,  ;  what  is  the  weight  of  the  whole  ? 

Ans.   i81b.  4oz.  3pwt. 

4.  In  35   pieces  of  cloth,  each  meafuring   27^  yards, 
how  many  yards  ?  Ans.  97  1£  yards. 

5.  How  much  brandy  in  9  cafks,  each  containing  45 
gallons,  3qt    ipt.  ?  Ans.  4izgal.  3qt.  ipt. 

6.  If  I  have  9  fields,  each  of  which  contains  1  2  acres, 
2  roods,  and  25  poles  ;  how  many  acres  are  there  in  the 
whole?  Ans.  113*0.  3r.  25p. 


9*  COMPOUND  DIVISION. 

COMPOUND  DIVISION 

IS  the  dividing  of  numbers  of  different  denominations  >\ 
in  doing  which,  always  begin  at  the  higheft,  and  when 
you  have  divided  that,  if  any  thing  remain,  reduce  it  to 
the  next  lower  denomination,  and  fo  on,  till  you  have  di- 
vided the  whole,  taking  care  to  fet  down  your  quotient 
figures  under  their  refpeclive  denominations. 

INTROUUCTORY  EXAMPLES. 

I.  2. 

£      s.    d.  D.    d.    c.  m, 

Divide  549     17     9  by  5  3)14     i     9     & 


Quotient 

£109     19     6i 

3. 

4> 

5- 

£     s- 

d.                  £      s. 

d. 

D.     c. 

m. 

4)73l     5 

ioi             6)37    -H 

4i 

7)25     49 

4 

6. 

7- 

T.  cwt. 

qrs.  Ib.   oz.   dr. 

Ib. 

oz.  pwt. 

gr. 

3)29     *3 

2       25        I        13 

10)849 

I  I        12 

H 

PRACTICAL  QUESTIONS. 

The  fame  note,  which  was  given  under  Practical  Quef- 
tions  in  Compound  Multiplication,  is  applicable  in  this 
place. 

CASE    1. 

Having  the  price  of  any  number  cf  yards,  &c.  within  the 
pence  table*  to  find  the  price  of  unity ^  or  \  yard :  If  the 
quantity  do  not  exceed  12,  proceed  by  fetting  down  the 
price,  and  dividing  it  by  the  quantity  ;  which  quotient 
will  be  the  price  of  one  yard,  required  j  but  if  the  quan- 
tity exceed  12,  then  divide  by  two  fuch  numbers,  as, 
when  multiplied  together,  will  produce  the  quantity,  and 
the  laft  quotient  will  be  the  value  of  i  yard, 


COMPOUND  DIVISION.  9? 

NOTE.     This  Cafe  proves  the  .firft  and  fecond  Cafes  iri 
Compound  Multiplication. 

EXAMPLES. 

1.  If  9  yards  of  cloth  coft  |44  3S*  7,   '  [what  is  it 

L^    3   93^c*  j 
per  yard  ? 

£.  s.     d.  D. 

9)4     3     7i  P)^^?* 

093!  Ans.  D.I -5485+  Aris. 

2.  If  7  ells  coft    {oY75'-^'}  what  coft  1  e11  ? 

A        Ci6s.  9^d. 
Ans-  [0.2-796 

3.  If  ii  fheep  coft  j -^    *s'  ^  *g        >  what  did  each 
=oft? 


is. 


4.  If  12  gallons  of  rum  coft 
is  it  per  gallon  ? 


Anc   /      T4S'    3i 

Ans'\D.2-386 


4.  If  84  cows  coft  i£2|3   130S^  }     what  is  the  price 

of  each  ?  .  3  os.  4|d. 

s' 


5.  If  132  buihels  of  corn  coft  |£2°g  I2S*  6d'|  what  is 
that  per  buihel  ?  A       f    35.  i^c& 


CASE    II. 

Having  the  price  of  a  hundred  weight  >  to  find  the  price  of 
lib  :   Divide  the  given  price  by  8,  that  quotient  by  7,  and 
this  quotient  by  2,  and  the  laft  quotient  will  be  the  price 
of  lib.  required. 


£4  COMPOUND  DIVISION. 

EXAMPLES. 

1.  If  icwt.  of  flax  coa  \  W  ?S*  8d*     1  what  is  that 

(  D.7  94c.  4m.  ^ 

per  pound  ? 

•£    s.    d.  D. 

8)2     7     8  Ii2)7'944(-O7i— 

784 

7)0     5  n£ 

104 

2)0    o  tod.  o£qr.  Ans.  70.  im,  nearly, 

2.  At    I  0  x  ICJjS     1  per  cwt.  what  coft  lib.  ? 


3,   At  {£62fS'}percwt.  what  coft  ilb.  ? 


At 


CASE      III. 

Having  the  price  of  fever  al  hundred  weight  >  to  Jind  the  price 
per  pound  :  Divide  the  whole  price  by  the  number  of  hun- 
dreds, which  will  give  the  price  per  cwt.  and  then  proceed 
as  in  the  lalt  Cafe. 

EXAMPLES. 


XA 
I.  If  5cwt.   of  fugar  coa 


that  perlb.  ? 

£.     s.   d. 

6     4 


is 


8)  2   13     8  price  of  icwt. 

7)  o     6     a|  price  of  14^.  or  -J-  cwt; 

2)  o    o  n^  price  of  alb.  or^Vcwt... 


COMPOUND  DIVISION.  9? 

H2lb.  in  a  cwt. 

5  cwt. 

'  *  c.m. 

5610)44-7  22  ('079-5-  —  ,  or  8  cents  nearly,  Ans* 
392 

552 
504 


482 

--  =J  nearly. 

560 

i.  If  Scwt.of  cocoa  cod   [ft* 
that  per  Ib.  ? 


£  I  what  is 

f    4'd. 
|jc.  ym.H- 


3.  If  siewt.offugarcoft  {£9  ''g^'}  what  is  it 
Perlb.?  r..d 

19C.+ 


4-  If  .Jew,,  of  cottonwool  coa 
what  is  that  per  Ib.  ?  f     8d. 


NOTE.  This  Cafe  proves  the  6th  in  Compound  Mul- 
tiplication. 

CASE   IV. 

Having  the  price  of  any  number  of  yards  *  &c.  to  fnd  the 
price  of  i  yard  :  Divide  the  price  by  the  quantity,  begin- 
ning at  the  higheft  denomination,  and,  if  any  thing  re- 
main, reduce  it  into  the  next,  and  everj  inferiour  denom- 
ination, and,  at  each  reduction,  divide  as  before,  remem- 
bering each  time  to  add  the  odd  (hillings,  pence,  £c.  if 
there  be  any,  and  you  will  have  the  value  of  unity  re- 
quired. 

NOTE.  If  there  be  ^,  1,  or  |  of  a  yard,  pound,  &c. 
multiply  both  the  price  and  quantity  by  4,  and  then 
proceed  as  above  directed  ;  or,  in  Federal  Money,  work 
by  decimals. 


96  COMPOUND  DIVISION. 

EXAMPLES. 

1.  If95ilb.   of  figsc^^D.j/^c^im.}^^   are 
they  per  Ib.  ? 

Ouantity=95JH>.                         Priced  r  6     13     6J 
Mul.  by       4 4^ 

Produces  ^82  for  a  divifor.       Producl  £66     14     3  for  a 

(dividend, 

382)66     14     3(0     3     5l!i§Perlb' 

20 
D.  c.m.dec.c.  m. 


382)1334(3  9 

1146  4775 

"Ts8  7843 

12  7640 

382)2259(5  2°37 

; 


349 
4 

382)1396(3 
1146 

250 

2.  If  147  bufhelsof  ryecoft 
itperbuftiel? 


3,  If  33i  yards  of  baUc  coft  {  J/^1  1^  }    what 

iS  it  per  yard  ?  Ans   J  »5S-  5id.  TTT 

s'  {D.2  570.  5m. 

NOTE.     This  proves  the  3d  and  4^  Cafes  in  Com 
pound  Multiplication  a 


COMPOUND  DIVISION  97 

PRACTICAL  QUESTIONS  IN  MONEY. 
,.  Divide  {^'\9;;f;m.}  among   5   men   and    4 

women,  and  give  the  men  twice  as  much  as  the  women. 
Men.  Worn. 

5  and  4 
Mult,  by  2 

10  {hares. 

Add 4  women's  (hares. 

14  the  number  of  equal  (hares  in  the  whole=Divifort 

£.      s.    d.  £.    s.    d. 

Divide  by  14)273     9    4(  19     *°    8=  i  woman's  fliare* 
14  4  women. 


*33 

126 

7 

20 

14)'49(10 
H 

9 

12 

lAhia'fS 

78 

2 

8=women's  (hare. 

19 

IO 

8 

2 

39 

1 

4=1  man's  (hare, 
5  men. 

195 

78 

6 

2 

8=men's  (hare. 
8=women's  Ihare- 

*7S 

9 

4  Proof. 

i  woman's 
4  women. 

26o*444=women's  ihare. 
65-111-1- 


i3O'222-l-=i  man's  Chare. 
5 


65  1  •  1  1  1+  =  men's  fiiarc. 
260-444+  =  women's  fhare. 
Proof. 


93  DUODECIMALS. 


,.  Divid,  a™n*   7   men  and  7 

women,  and  give  the  women  3  times  f«  much  as  the  men. 


Ans. 

>  =a  woman's  {hare. 

3.  Divide  j  J^  j  »2  *7c  I  amonS  4  men»  6  women, 
and  9  boys  :  Give  each  man  double  to  a  woman,  and 
each  woman  double  to  a  boy. 


Ans. 

I 


"2l.    2S.    lOd.   7  i      n 

j-          c >  =  a  woman's  mare. 

4!.    s"s.   8d.  5    n 

.D.I4  28c._|  =  amans{hare. 


DUODECIMALS  : 

0*  CROSS  MULTIPLICATION, 

IS  a  Rule,  made  ufe  of  bv  workmen  and  artificers  ia 
cafting  up  the  contents  of  their  works. 

Dimenfions  are  generally  taken  in  feet,  inches,  and 
parts. 

Inches  and  parts  are  fometimes  called  primes,  feconds, 
thirds,  &c.  and  are  marked  thus  :  inches  or  primes,  ('), 
feconds  ("),  thirds  ('"),  fourths  (""),  &c. 

This  method  of  multiplying  is  not  confined  to  tivtlves, 
but  may  be  greatly  extended  :  For  any  number,  whether 
its  inferiour  denominations  decreafe  from  the  integer  in 
the  fame  ratio,  or  not,  may  be  multiplied  crofs- wife  ; 
and  for  the  better  underftanding  of  it,  the  learner  muft 
obferve,  that  if  he  multiplies  any  denomination  by  an  in- 
teger, the  value  of  an  unit  in  the  product  will  be  equal  to 
the  value  of  an  unit  in  the  multiplicand  ;  but  if  he  multi- 
plies by  any  number  of  an  inferiour  denomination,  the 
of  an  unit  in  the  product  will  be,  &  much  inferiour 


DUODECIMALS.  99 

to  the  value  of  an  unit  in  the  multiplicand  as  an  unit  of 
the  multiplier  is  lefs  than  an  integer. 

Thus,  pounds,  multiplied  by  pounds,  are  pounds  ; 
pounds,  multiplied  by  fliiliings,  are  (hillings,  &c.  ;  shil- 
lings, multiplied  by  fhillings,  are  twentieths  of  a  milling  ; 
shillings,  multiplied  by  pence,  are  twentieths  of  a  penny  ; 
pence,  multiplied  by  pence,  are  24Oths  of  a  penny,  &c. 
RULE. 

1.  Under  the  multiplicand  write  the  correfponding  de- 
nominations of  the  multiplier. 

2.  Multiply  each  term  in  the   multiplicand,  beginning 
at  the  loweft,    by  the  highefl  denomination   in  the  rniu .-.- 
plier,    and   write    the   refill t   of  each  under  its  refpecYive 
term,  obferving,  in  duodecimals,  to  carry  an  unit  for  ev- 
ery   12,  from  each  lower  denomination  to  its  next  fuperi- 
our?  and  for  other  numbers  accordingly. 

3>  In  the  fame  manner  multiply  all  the  multiplicand 
by  the  primes,  or  fecond  denomination  in  the  multiplier, 
and  fet  the  refult  of  each  term  one  place  removed  to  the 
right  hand  of  thofe  in  the  multiplicand. 

4  Do  the  fame  with  the  feconds  in  the  multiplier,  fct- 
ting  the  refult  of  each  term  two  places  to  the  right  hand 
hand  of  thofe  in  the  multiplicand. 

5.  Proceed  in  like  manner  with  all  the  reft  of  the  de- 
nominations, and  their  fum  will  be  the  anfwer  required, 

EXAMPLES. 
j.  Multiply  2^  feet  by  2 ^  feet. 

F.    '  Or  thus, 

a-    6-  2i' 

26  2^ 

50  5 

i     3     o  ij 


Ans.  6     3  Ans.  6£  fquare  feet  =  6  feet 

36  inches.      So  that  the  3  is  not  3  inches* 
but  36in.  or  £  of  a  fquare  foot. 
Or  thus.     2-5 


5° 
Ans.  6  25  fquare  feet.. 


ioo  RULE  OF  THREE  DIRECT. 

a.  Multiply  9f.  8'  6"  by  76  9'  3". 
F.     '    " 
986 
793 


6711     6  =Prod.  by  the  feet  in  the  multi  plier, 

734  6V        =do.  by  the  primes. 

2     5  i     6"''  =do.  by  the  feconds. 

75     5     3  7     6  Anfwer. 


3-  How  many  fquare  feet  in  a  board  17  feet  6  inches 
Jong,  and  i  foot  5  inches  wide  ?  Ans.  24ft.  10'  it" 

4.  How  many  cubick  feet  in  a  ftick  of  timber,  1 2  feet 
10  inches  long,  i  foot  7  inches  wide,  and  i  foot  9  inches 
thick  ?  Ans.   35ft.  6'  8"  6'' 

5.  Multiply  3!.  6s.  8d.  by  2!.  53.  7  4* 

£  s.  d. 
368 
257 

£3X/2=£6      =  600 

6s.X^*2=l2S.        =    O    12       O 

8d.X^2=i6d.  —014 

155.  =  o  15     o 

£«.  =  o     i     6 

^^d.  =00     2 

2  id.  =o     i     9 

6s"x7d.=4£d.  =00     2T^ 

:^nd.  =   O      O 


Ans.  7 


SINGLE  RULE  OF  THREE  DIRECT. 

THE  Rule  of  Three  DirecT:  teacheth,  by  having  three 
numbers  given,  to  find  a  fourth,  that  ihall  have  the  fame 
proportion  to  the  third,  as  the  fecond  hath  to  the  firft, 


RULE  OF  THREE  DIRECT.  lot 

If  more  require  more,  or  iefs,  require  Itfs,  the  quefticm 
belongs  to  the  Rule  of  Three  Direct. 

But  if  more  require  lefs.  or  lefs  require  more,  it  belongs 
to  the  Rule  of  Three  Inverfe.* 

RULE  f 

i.  State  the  queftion  by  making  that  number, 
which  afks  the  queftion,  the  third  term,  or  putting  h" 
in  the  third  place  :  That,  which  is  of  the  fame  name 
or  quality  as  the  demand,  the  firit  term  ;  and  that  which 
is  of  the  fame  name  or  quality  with  the  anfwer  required^ 
the  fecond  term. 

•  More  requiring  mjrt,  is  when  the  third  term  is  greater  than  the 
firft,  and  requires  the  fourth  term  to  be  greater  than  the  fero  id. 
And  lefs  requiring  /<•/},  is,  when  the  third  terra  is  Itfs  than  the  firft, 
requires  the  fourth  term  to  be  lefs  than  the  fecond 

Alfo,  mire  requiring  lefs,  is  when  the  third  term  is  greater  than  the 
fir  ft.  and  requires  the  fourth  term  to  be  lefs  than  the  fecond  And 
lefs  requiring  more  is  when  the  th;rd  term  is  lef>  than  the  fir  ft,  and  re- 
quires the  fourth  term  to  be  greater  than  the  fecoud. 

f  This  Rule,  on  account  of  its  great  and  extenfive  uftfulnefs,  is 
fometim  s  called  the  Golden  Rule  of  Proportion  :  for,  on  a  proper  ap- 
plication of  it  and  the  preceding  rules,  the  whole  bufmefs  of  Arith.ne- 
tick,as  well  as  every  mathematical  inquiry,  depends  The  rule  itfelt* 
is  founded  on  this  obvious  principle,  that  the  magnitude  or  quantity 
of  any  cffcdl  varies  conflantly  in  proportion  to  the  varying  p«rt  oi  the 
caufc  :  Thus  thequmtify  of  goods*  bought,  i»  in  proportion  to  the 
money  laid  i>ut ;— the  fpace,  gone  over  by  an  uniform  motion,  is  in 
proportion  to  the  time,  &c 

It  has  been  ihovvn  in  Multiplication  of  Money  that  the  price  of 
one  multipled  by  the  quantity,  is  the  price  of  the  whole  ;  and  in 
Divifion,  tint  the  price  of  the  whole  divided  by  the  quantity  is  ihe 
price  of  one:  Now,  in  all  cafes  of  valuing  goods,  &c.  where  one  is 
the  firft  term  of  the  proportion,  it  is  plain  thar  the  anfwer,  found  by 
this  rule,  will  be  the  fame  as  that  found  by  multiplied  >n  of  money, 
and  where  one  is  the  laft  term  of  the  proportion,  it  will  be  the  fame 
as  that,  found  by  Divifion  of  Money. 

In  like  ma-intr,  if  the  firft  term  be  any  number  whatever,  it  is  p!  tia 
that  the  product  of  the  ietond  and  third  ternis  will  he  greater  than 
the  true  anfwer  required,  by  as  much  as  the  price  in  the  ftrond  term 
exceeds  the  price  of  one,  or  as  the  firft  term  f  xceeds  an  unit  ;  coafc- 
qu-ntly  this  proJucl,  divided  by  the  lirft  term,  will  give  the  true  an 
iwer  required 

No'f  i    When  it  can  be  done,  multiply  and  divide  as  in  Compour 
Multiplication, and  Compound  Diviuun. 

1  ^ 


102  RULE  OF  THREE  DIRECT. 

2  Multiply  the  fecond  and  third  numbers  together  . 
divide  the  product  by  the  firft,  and  the  quotient  will  be 
the  anfwer  to  the  question,  which  (as  alfo  the  remainder) 
will  be  in  the  fame  denomination  you  left  the  fecond  term 

1.  If  the  firft  term,  and  cither  the  fecond  or  third  can  be  divided 
by  any  number  without  a  remainder,  let  them  be  divided,  and  the 
quotient  ufed  inftead  of  them. 

The  following  methods  of  operation,  when  they  can  be  ufcd,  per- 
form the  work  in  a  much  fhorter  manner  than  the  general  rule. 

1.  Divide  the  fecond  term  by  the  firft  ;  multiply  the  quotient  in- 
to the  third,  and  the  product  will  be  the  anfwer. 

2.  Divide  the  third  term   by  the  firft  ;  multiply  the    quotient  into 
the  fecond,  and  the  product  will  be  the  anfwer. 

3.  Divide  the  firft  term  by  the  fecond,  and  the  third  by  that  quo- 
tient and  the  laft  quotient  will  be  the  anfwer. 

4.  D;vide  ihe  firft  term  by  the  third,  and  the  ftcond  by  that  quo- 
tient, and  the  laft  quotient  will  be  the  anfwer 

Note.  The  term  whichafksor  moves  the  queftion,  has  generally 
fome  words  like  thefe  before  it.  viz.  What  will  ?  What  coft  ?  How 
many  ?  How  far  r  How  long  ?  How  much  ?  &c. 

OBSERVATIONS. 

I.  The  fourth  number  is  always  found  in  the  name  in  which  the 
fecond  is  given,  or  reduced  to  ;  which,  if  it  be  not  the  higlieft  de- 
nomination of  its  kind,  reduce  to  the  higheft,  whea  it  can  be  clone. 

a.  When  the  fecond  number  is  of  divers  denominations,  bring  it 
to  the  loweft  mentioned,  aud  :he  fourth  will  be  found  in  the  fame 
name  to  which  the  fecond  is  reduced,  which  reduce  back  to  the 
higheft  poffi'jle. 

3.  If  the  firft  and  third  be  of  different    names,  or  one  or  both  of 
'divers  denominations,  reduce  them  both  to  the  loweft  dcnominatioa 
mentioned  in  either. 

4.  When  the  product  of  the  fecond  and  third  is  divided  by  the 
fitft  ;  if  there  be  a  remainder  after  the   divifion,  and  the  quotient   be 
not  the  leaft  denomination  of  its  kind  ;  then  multiply  the  remainder  by 
that  number,  which  one  of  the  fame  denomination  with  the  quotient 
contains  of  the  next  lefs,  and   divide  this  product  again   by  the  firft 
number ;  and  thus  proceed,  till  the  lea  ft  denomination  be  found,  or  till 
Bothing  rentain 

5.  If  the  firft  number  be  greater  than  the  product  of  the  fecond 
and  tlvrd  ;  then  bring  the  fecond  to  a  lower  denomination. 

6.  When   any  number    of  barrels,  bales,  or  other  packages    or 
pieces  are  given,  each  containing  an  equal  quantity,  let  the    content 
of  one  be  i  educed  to  the  iowcft  name,  and   then    multiplied    by  the 
given  number   of  packages  or  pieces 

7.  If  the  given  barrels,  bales,  pieces,  &c.  be  of  unequal   contents, 
(as  it  moft  generally  happens)  put  the  feparate  content  of  each  pro- 
.J>er'y  under  one  another,  then  add  th«n  together,  and  you  will 

the  \vhole  quantity. 


RULE  OF  THREE  DIRECT.  103: 

in,  and  which  may  be  brought  into  any  other  denomination 
required. 

Two  or  more  Ratings,  are  fometimes  neceflary,  which 
may  always  be  known  from  the  nature  of  the  queftion. 

The  method  of  Proof  is  by  inverting  the  queftion. 

EXAMPLES. 

T.  If  61b.  of  fugar  coft  j^*'         |  what    will 

coft  at  the  fame  rate  ? 

Ib.    s.      Ib.  Ib.  D.  c.       Ib. 

As  6  :  9  ::  30  :  the  Ans.  As  6  :  i    50  ::  30  :  the  Ans, 
9  30 

6)270  6)45-00 

45s.=>£2  55.  Ans.        D-7  50  Ans. 

Here  the  firft  claufe  (if  6lb.  of  fugar  coft  95.  orD.i 
5oc.)  fuppofesthe  rate  ;  then  follows  the  queftion  :  What 
will  3olb.  coft  ?  3olb.  which  moves  the  queftion,  is  the 
3d  term.  61b.  the  fame  kind,  is  the  ift,  and  95.  (or  D.I 
5oc.)  the  2d. 

Again.  By  inverting  the  order  of  the  queftion,  it  will  be, 

.  If  -J  j^S*        i    buy  61b.  of  fugar,  how  much  will 


s.     Ib.       s. 

As  9  :  6  ::  4$  :  the  anfwer. 
6 

9)270 

3olb.  Ans. 

D.  c.      Ib.    D.  c. 

As    i   50  :  6  ::  7  50  :  the  anfwer. 
6 

oo(3olb.  aniwsr, 
45 


104  RULE  OF  THREE  DIRECT. 

Again,  3.   If  3olb.  of  fugar  be  worth  •<   .«  2  ^s*      I 

{V  I         J  Js 

n^S         f   - 
-u.i  500.3 

s,       Ib.       s. 

As  45  :   30  ::  9  :  the  answer. 
9 

45)270(61*0.  the  anfwer. 

270 
--  D.         Ib.     D. 

As  7-50  :  30  ::  1-50  :  the  anfwer. 

30 

7'5)45'0o(61b,  anfwer. 
45  o 

Again,  4.    Suppofe  j^5**^  j  will  buy  3olb.  of  fu*- 

gar,  what  will  61b    of  the  lame  fu^ar  coft  ? 

Ib.       s.       Ib. 

As  30  :  45  ::  6  :  the  anfwer. 
6 


anfwer. 

Ib.     D.          Ib. 

As  30  :  7-50  ::  6  :  the  anfwen 
6 


D  i  '50  anfwer. 

N.  B.  The  three  laft  queftions  are  only  the  firft  varied, 
being  put  merely  to  fh-jw  how  any  queilion,  in  this  Kiue, 
may  be  inverted. 

5.  If  5yds.  cloth  coft  |  *^  Ios'  I  what  will  20  yards 
ditto  come  to  ? 


RULE  Of  THREE  DIRECT. 

yds. 

As  5     : 
3.0    -f 

2|o)i2|os.=;£6  the  anfwer. 

Here  I  divide  the  2-i  term  by  the   i  ft,  and  multiply  the 
Quotient  into  the  3d  for  the  anfwer. 

yds.       s.  yds. 

Again,  6.          As  5     :     30     ::     20 
20-5    =      4 

f2 os.  =£6  anfwer. 

tlere  I  divide  the  3d  term  by  the  ift,  and  multiply  th&. 
quoiient  into  the  2d,  for  the  anfwer. 

Thefe  operations  would  be,  perhaps,  ftill  more  apparent., 
if  performed  in  Federal  Money.     Thus : 

yds.    D.     yds.  yds.  D.     yds. 

As  5  :  5  ::   20  As     5   :  5  ::  20 

54-J=     i  20-^5  =  4 

D.2o  anfwer.  D.2O  anfwer. 

7.  If  20  yards  coft  D.i2o,  how   many   yards  may  X 
have  for  Djo  ? 

D.  yds.        D. 

As   120     :     20     ::     30 

I2o-r-2o=6  quotient,  and  30-7-6=5  yards,  anfwer. 
Here  I  divide  the  ift  term  by  the  zd,  and  then;  the  35! 
term  by  the  quotient  for  the  anfwer. 

D.         yds.         D. 
Again,  8.     As   120     :     20     ::     30 
120-7-30=^  quotient,  and  20-7-4=5  yards,  anfwer, 
Here  I  divide  the  ift  term  by  the  3d,  and  then,  the  2d 
term  by  that  quotient  for  the  anfwer. 

9.  If  i cwt.  of  tobacco  coft  £$   12  9  J,  what  will  8cwt. 
diuo  coft  ? 

cwt.       £    s.  d.  cwt. 

As   I      :     5    12  9^     ::     8 
8 


s.  £45     ^  4 

Here  there  is  no  need  of  reducing  the  middle  term, 
becaufe  it  can  be  performed  by  compound  multiplication^ 
the  6rft  term  being  an  unit, 


RULE  OF  THREE  DIRECT. 

10.  If  8cwt.  of  tobacco  coft  £45  2  4,  what  is  that 
per  cwt.  ? 

£      s-    d- 
8)45     2     4 
Ans.     5   12     9^ 

Here  there  is  no  need  of  reducing  the  middle  term,  be- 
caufe  It  may  be  performed  by  compound  divifion  only,  the 
3d  term  being  an  unit. 

f/zf   i7s    6d    1 
^i.  If  pew.  3qr..  fugar  eoft  |*  ^  ^  ?m  j  what 

will  2 cwt.  iqr.  i  jib.  coft  ? 

Cwt.  qrs.  £    s.  d.                         Cwt  qr.lb. 

93  27   17  6                          21   ii 

4  20                                  4 

39  557  9 

28  12  28 

312  6690  73 

78  19 

263 
the  anfwer. 


i:C92 

Ib. 
As  109* 

d. 

:     6690     :: 
26? 

2007 
4014 
1338 

WWM^WMMMM^V  If   ^ 

Ib. 
263 

3d. 


RULE  OF  THREE  DIRECT.  107 

NOTE  i.  If  you  look  at  the  dating,  you  will  fee  that 
the  firft  and  third  terms  are  of  the  fame  kind,  but  of  dif- 
ferent denominations,  and  therefore  are  reduced  to  the 
fame  name  or  denomination,  and  that  the  demand  of  the 
queftion  lies  on  the  3d  term. 

2.  That  the  middle  term,  being  given  in  pounds,  fliill- 
ings  and  pence,  is  reduced  to  pence.     But 

3.  If  the  fecond  term  were  in  federal  money  it  would  be 
fufficient  to  proceed  according  to  decimals.     Thus  : 

Ib.        D.  lb. 

As    1092  :  92-917  ::  263  :  the  anfwer. 


278751 
557*502 
185834 

--  -  D.c.m. 
1092)24437-1  71  (22'378-f,  anfwer. 
2184 


8736 
395 

12.  If  57  yards  coa    {oo      what  wil1  9  yafds 
coil  at  that  rate  ? 

yds.        £        yds. 
As  57     :     69     ::     9 
9 

57)621(10!.  anfwer. 
57 

5* 

20 

1020        Carried  over- 


io8          RULE  OF  THREE  DIRECT. 

Brought  over.        57)1020(175. 
57 

450 
399 


57)61  2(iod. 
57 

42 
4 


114 

Here,  all  the  terms  being  whole  numbers,  there  is  no 
need  of  reducing  the  middle  one  until  after  ftating. 
l*he  fame  in  Federal  money  would  ftand  thus  ; 

yds.         D  •         yds. 
As  57     :     230     ::     9     :     the  anfwer. 
9 

57)2070(360.  3  ic.  5i4m.  AnfwerB 
171 

""360" 
34* 

180 
171 

90 
57 

330 


57     '9 


RULE  OF  THREE  DIRECT.  109 

13.  If  my  income  be  109  guineas  per  annum,  I  cefire 
to  know  what  I  may  fpend  per  day,  fo  that  I  may  lay  up 
45!.  at  the  year's  end  ?  Ans.  55.   icjd.  -^T  per  day- 

NOTE    i.  You  mull  fubtraft  45!.  from  the  value  of  109 
guineas. 

2.  There  being  365  days  in  a  year,  your  qusdion  mud 
next  be  ftated  thus  : 

D.     Guin.  £      D.     s.  d.  qr. 
As  365  :   109—45  ::   i   15    10  3 T£T  the  Ans: 

14.  If  my  falary   be  43!.  us.   jd.    per  annum,   what 
does  it  amount  to  per  week  ?  Ans.    163.  9}Jd. 

The  Stating.  W.     £    s.    d.     W. 

As  52   :  43    12   5   ::   i   :  the  anfwer. 
NOTE.     As  there  are  52  weeks  and  i    day  in   a  year, 
you  will  get  the  true  anfwer  to  the  above  queftion  by  the 
following  ratio. 

D.       £     s.  d.     D. 
As  365   :  43    12  5   ::  7   :    i6s.   8|-|Jd. 

15.  Suppofe  my  income  to  be  i6s.  8£|4d.    per  week, 
what  is  it  per  annum  ?  Ans.  43!.   135.  y£d.  -/$•£• 

The  Stating. 

D.     s.    d.  D. 

As  7  :   1 6  8f|4  ::  365  :  43!.   125.  jd.  Ans. 

16.  If  I  am  to  pay  is.  7d.  per  week  for  pafturing   a 
cow  j  what  muft  I  give  per  week  for  37  cows  ? 

C.      s.  d.       C. 
As  i   :   i   7  ::  37  :  2!.  i8s.  7d.  Ans. 

1 7.  How  many  yards  of  cloth  may  be  bought  for  D.  1 95 
75c.  of  which  9?yds.  coft  udol.  2c.  ? 

D.  c.      yds.      D.    c.       yds.  qrs. 
As   ii  02   :  9t  ::  195  75  :    168  3  Ans. 

1 8.  If  I  buy   57  yards  of  cloth  for  49  guineas  ;  what 
did  it  coft  per  ell  Englifh  ? 

yds.  guin.  yds. 
As  57  :  49  ::   i^  il.  TOS.  i?VTd.  Ans. 

19.  A  merchant,  failing  in  trade,   owes  in  all  ^3475^ 
and  has  in  money  and  effects  but  £2316  134:  Now  iup- 
pofing  his    effects  are  delivered  up,  pray  what  will  each 
creditor  receive  on  the  pound  ? 

£•      £.    ••  ^  £• 

As  3475  :  2316  13  4  ;:  i  :  £3  133.  4d.  Ans, 
K. 


1  ID  1UJLE  OF  THREE  DIRECT. 

20.  A  owes  B   3475!.  but  B  compounds  with  him  for 
a  35.  4<1.  on  the  pound:    Pray  what  mull  he  receive  for 
his  debt  ? 

£      s-  d.         £ 
As   i   :    13  4  ::   3475   :   2316!.   135.  4d.  Ans. 

21.  If  the    diftance  from   Newburyport  to  York  be  3  T 
miles  j    1   demand  how   many  times  a  wheel,  whofe  cir- 
cumference is   15^  feet  will  turn  round  in  performing  the 
journey  ? 

Feet.  Cir.     M.        Cir. 
As  15-3-  :   i   ::    31   :    10560   times,  Anfwer. 

22.  Bought  9  chefts  of  tea,  each  weighing  3cwt.  2qrs. 

2  lib.  at  ^1.  95.  per  cwt.  ;  what  came  they  to  ? 

Cwt.    £  s.      C.  qr.ib 
As  i   :  4  9  ::  3   2   21X9  :  147!.  135.8^. 

23.  What  will  37^grofsof  buttons  come  to  at  13  cents 
j»er  dozen  ? 

Doz.  c.   '  Grofs.     D.  c. 
As  i    :   13    ::  37^    :  58  50    Ans. 

24.  A  farm,  containing    i25ac.  3r.  27p.  is  rented   at 
D.I  i  5oc.  per  acre  ;  what  is  the  yearly  rent  of  that  farm  ? 

ac.     D.  c.     -  ac.  r.  p.         D.    c.m. 
As  i    :   ii  50  ::   125  3    27  :   1447  6  5 £  Ans. 

25.  If  a  fhip  coft  537!.  what  are  |-  of  her  worth  ? 

Eight.     £     Eight. 

As  8  :  537  ::  3  :  201!.  75.  6d.  Ans. 

26.  If  y^  of  a  fhip  coft  i  i  630.  what  is  the  whole  worth  ? 

Sixt.       D.      Sixt.      D.     c.  m. 
As  7  :   1163  ::   16  :  2658  28  5  Ans. 

27.  Bought  a  cafli  of  w»ne  at  76c.  5m.  per  gallon,  for 
125  dollars";  how  much  did  it  contain  ; 

Ans.   i63gal.   iqt.  iT3Tpt» 

28.  What  comes  the  iafurance  of  537!.  153.  to  at  4^!. 
per  centum  ? 

£     £     £    s.    £  s.  a. 

As  ico  :  4!  ::  537  15  :  24  2   u^  TV  Ans. 

29.  What  come   the  commiffions  of  785!.  to  at  3^ 
guineas  per  cent.  ?  Ans.    38!.  pi,  3^-  T4Dd. 


RULE  OF  THREE  DIRECT.  in 

30.  A  merchant  bought  9  packages  of  cloth,  at  3  guin- 
eas for  7  yards  :  each  package  contained  8  parcels,  each 
parcel,  12  pieces,  and  each  piece,  20  yards  :  how  many 
dollars  came  the  whole  to,  and  how  many  per  yard  ? 

Yds.  guin.  pack.      D. 
As  7   :  3  ::  9  :  34560  Ans,  for  the  whole  coft. 

Yds.  guin.  yd.  D. 
As  7   :   3  ::    1:2   Ans.  per  yard. 


31.  A  merchant  bought  49  tuns  of  wine  for 
freight  coft  0.90  ;    duties,  040;    cellar,  D.JI   670  ; 
oihsr  charges  0.50,  and  he  would  gain  U.  18-5  by  die  bar- 
gain ;  what  mail  I  give  him  for  23  tuns  ? 

Tuns.  D.    D.   D.    D.c.     D.     Tuns.  D. 
As  49  :  910+90+40+37  67+185  ::  23  :  613  330.  Ans. 

32.  If  D.  100  gain  D.6  in  a  year,  what  will  D  475  gain 
ia  that  time  ?  Ans.  D  28  500. 

33.  Ths  earrh  being  360  degrees  in  circumference,  turns 
round  on  its  axis  in  24  hours  ;  how  far  does  it  turn  in  one 
minute,  in  the  43d  parallel  of  latitude  ;  the  degree  of  Ion- 
gitude,  in  this  latitude,  being  about  5  1  ftatute  miles  ? 

H.      D.     M.     M.      M. 
As  24  :   360X51   ::    i   :   12^  Ans. 

34.  Shipt  for  the  Weft   Indies  225    quintals  of  Jifh,  at 
?5s.  6d.  per  quintal  j  37003  feet  of  boards,  at  S^doll,  per 
I  coo  ;   1-2000  fhingles,  at  Jguin.  per  JOOG;   19000  hoops, 
at  i^doll.  per  1000,  and  53  half  joes  ;  and  in  return,  I 
have   h.id    jooogals.  of  rum,   at    is.   3<i.    per   gallon  ; 
2700  gallons  of  molaties  at  ^d.  per  gallon  ;    ijoolb.  of 
coffee,  at  8Ji.  per  Ib.  ;  and  i^C-vt.  of  fugar,  at  123.   3d. 
per  cwt.  and  my  charges   on  the  voyage  were  37!.   i  2s.  ; 
Pray,  did  I  gain  or  loie,  and  how  much,  by  the  voyage  ? 

Ans.   loft  £134  9  9« 

3  5.  If  a  (lafF,  4  feet  long,  cad  a  fliads  (on  level  ground  ) 
7  fest  ;  what  i:>  the  height  of  that  itceple,  whofe  ihad;,  at 
the  fame  time,  mcaHires  198  feet  ? 

F.  ih,     F.  hci.     F.  (h.      F.  hei. 
As  7     :     4     ;:     198     :     iij^Ans.. 


ii2  RULE  OF  THREE  DIRECT. 

*36.  Suppofe  a  tax  of  D  755  be  laid  on  a  town,  and  the 
inventory  of  all  the  eftates  in  the  town  amounts  to  9345^ 
dollars,  what  mud  A  pay,  whofe  eftate  is  D.  149  ? 

D.         D.  D-      D.  c.  m. 

As  9345  :  755  "   '49  :   12   I2  7  Ans. 

*  It  may  net  he  amifs  to  fho'.v  the  genera!  method  of aflcffing  town 
or  parifli  taxes.  Firft,  then,  an  inventory  of  the  value  of  all  the 
tftates,  both  real  and  perfonal,  and  the  number  of  p<  1's,  for  whicti 
each  perfon  is  rateable;  muft  be  taken  in  feparate  columns;  The  mofl 
concife  way  is  then  to  make  the  total  value  of  the  inventory  the  firft 
term,  the  tax  to  be  sflefled,  the  fecond,  andD.i  the  third',  and  the 
quotient  will  (how  the  value  on  the  dollar  :  adly,  Make  a  table, 
by  multiplying  the  value  on  the  dollar  by  i,  2,  3,  4,  &c.— 3dly, 
From  the  inventory  take  the  real  and  perfonal  eftatea  of  each  man, 
and  find  them  feparately  in  the  table,  which  will  {hew  you  each 
man's  proporrional  {hare  of  the  tax  for  real  and  perfonal  eftates. 

Note.  If  any  part  of  the  tax  is  averaged  on  the  polls,  or  otherwife, 
before  fhting  to  find  the  value  on  the  dollar,  you  rnuft  deducfl  the 
fum  of  the  average  tax  from  the  whole  fum  to  be  aflefled :  for  which 
average,  you  muft  have  a  feparate  column,  as  well  as  for  the  real 
and  perfonal  eftates. 

EXAMPLE. 

Snppofe  the  General  Court  fliould  grant  a  tax  of  0.500000,  of 
which  the  town  of  Ncwburyport  is  to  pay  D  53  M  500.  and,  of  which 
the  polls,  being  1550,  are  to  pay  D  i  ajc.  each  . — The  town's  inven- 
tory amounts  to  0.450000;  what  will  it  be  on  the  dollar,  and  what 
is  A's  tax,  whofe  eftate,  (as  by  the  inventory)  is  as  follows,  viz.  real 
1X1376,  perfonal  0.1149,  and  he  has  3  poll's  ? 
Pol.  D.  c.  Pol.  D.  c. 

Firft,  As  i  :   i   25  ::   1550  :   1937  50  the   average  part  of  the 
tax  to  be  deduced  from  0.5312  500.  and  there  will  remain  D.J37J. 
D.  D.         D. 

Secondly,  A?  45000  :   3375  ::    i   :  7^01.  on  the  dollar. 

TABLE. 
D.        D.  c.  m.  D.          D    c.  m.  D.         D.    c. 

1  is   o  o  75  20  is   o  -15  o  200  is   i  50 

2  —  015  30  —  0225  300  —  2    25 

3  —  o  2  2^     40  —  o  30  o      400  —  3  oo 
i  —  o  3  o      50  —  o  37  5      500  —  3  75 

5  —  °  3  ?i  60  —  c  45  o  600  —  4  50 

6  —  045  70  —  0525  700  —  5  25 

7  — •  o  5  2^  80  —  o  60  o  800  —  6  oo 

8  —  060  90  —  0675  9°°  —  ^  75 

9  —  o  6  7 g-  loo  —  o  75  o  1000  —  7  50 
Jo  —  o  7  5 

N  >w,  to  find  what  A's  rate  will  be    . 


RULE  OF  THREE  DIRECT.        -  113 

37.  If  52  gallons  of  water  in  one  hour,  fill  into  a  cif- 
tern,  containing  230  gallons,  and  by  a  pipe  in  the  ciftern 
3  5  gallons  run  out  in   an  hour  ;  in  what  time  will  h  be 
filled? 

Gal    gal.    h.      gal.      h. 
As  50 — 35  :   £   ::  230  :   15.5-  Ans. 

38.  A  butcher  went  with  ^4: 6,  to  buy  cattle  :    Oxena 
at  £22  each,  cows  at  £4,  fteers  at  /.*3    IDS.  and  calves  at 
£2    »os.  and  of  each  a  like  number  ;  how  many  of  each 
could  fee  purchafe  with  that  Aim  ? 

£    £  £    s    £    s-    eacn-     £        each. 
As  22+4+3    *o+2   jo  s   i   ::  4'6  :   13   Ans. 

3.9.  Said  Harry  to  D^ck,  my  purfe  and  money  arc 
worth  3^  guineas  but  the  money  is  worth  eleven  times  as 
much  as  the  purfe  ;  pray,  how  much  money  is  there  in 
it  ?  Guin.  s  d. 

As  12:1    ::    3i    :    7    7 
Then  £4  us. — 73.  yd.^j.  33.  fd     Ans, 

40.  How  many  dozen  pair  of  gloves,  at  1 3  groats  per 
pair,  may  I  have  for  125  dollars  ? 

Gr.    pr.      dol.    d<  z.  pr. 
As   13  :   i   ::   125   :    14  5^  An?. 

41.  There  is  a  cidern,  having  four  cocks  ;  the  firfl  will 
empty  it  in  ten  minutes  ;  the  fecond  in    10  minutes  ;  ,the 
third  in  40,  and  the  fourth  in  80  minutes  j  in  what  time 
will  all  four  running  together,  empty  it  ,? 

His  real  eflate  being  D  1376,  I  find,  by  the  table, 
That  D.I  coo       is         D.;   500. 
That  D.  300   is  a  25 

That  70  is  j»     5m. 

And  that  6  is  45 

F  >r  his  real  cftate,  D  10  31 

In  like  manner,  I  find  his  tax")  ,^o   ,     >  i 
fc>"  pcrf-mal  efhte  to  be  3  '^     >  -f  D  jo  &  = 

Hu  tlirte  polls,  at  f)  i  Z5c.  e^ch  are  3   75 
or  D.^^  6c   Ans. 


K-.il.  IPerfonal.)    Pills. 
L)  c.  m     O    c.  m  |D     c.   m 


o|  8  61  7^1  3  75 
a 


Total 
D   c    -n. 


ji4  RULE  OF  THREE  BIRECT 


Gift.     Min.  Gift.  Min. 
As   uj  :  60  ::    i   :  5^  Ans. 

42.  A  and   B  depart  from  the  fame  place,  and  travel 
the  fame  road  :  but  A  goes  5  days  before  B,  at  the  rate  of 
20  miles  per  day  ;  B  follows  at  the    rate  of  25  miles  per 
day  :  In  what  time  and  diftance  will  he  overtake  A  ? 

M.     M.  D.     M.  D.  D.  D.    M      D.      M. 

As  25  —  20  :  i  ::  20X5  :  20.     And,  As  i  :  25  ::  20  :  500 

43.  Sold  a  cargo  of  flaxfeed  in  Ireland,  for  /,  i  795  IDS. 
Irilh  money  ;  what  does  that  amount  to,  in  M  diachufetts 
currency,  £8  1  55.  Irifh  being  equal  to  £  i  oo  MuilUchufetts  f 

Irifh.       Maff.          Irith.         M*fT. 
As  £81-^  :  £100  ::  £i  795^  :  £2209  )6s.  i  id.  Ans. 
Or,  As  £13  :  £1795?  ::  £l6    :     £2209  •.  6s.  i  id.  as  be- 
fore, becaufe    £13  Inih   are  equal  to  £16   Maffachufetts. 

44.  My  correfpondent  in  Maryland  ptirchafed  a  cargo 
of  flour  for  me,  for  437!.  that  currency  ;  how  much  M  if- 
fachufetts  money  mult  I    remit  him,  125!.  Maryland   be- 
ing equal  to  100!.  MafTachufetts,  or  5  Mar  =4  Mail  ? 

Ans.    349!.  i2s,  MafT. 

45.  A  bill  of  exchange  was  accepted  at  Newburyport 
for  the  payment  of  345!.  IDS.  for  the  like  value  delivered 
m  New-  York,  at     13  3^!.    New  York  currency  for    iool, 
Maffachufetts  ditto  ;  how  much    money  was  paid  in  New- 
York  ? 

Maff.     N.  Y.       Maff.  N.  Y. 

As  75!.  :    iool.  :;  345!.  los.  :  460).  135.  4<L  Ans. 

46.  When  the  exchange  from  M  iiTichufetu  to  Georgia 
is    83^1.    Georgia  per    loci.   Maflachufetts,    how   much 
Maffachufetts  money  mult  be  paid   in  Bofton  to  balance 
457!.  Georgia  currency  ?  Ans  5481.  Sb   Miff. 

47.  A  merchant  delivered  at  Bofton,   320!.  MaflUchu- 
fetts  currency,   to  receive   400!    in    Philadelphia  j  what 
was  the  Maffachufetts  pound  valued  at  ? 

Ans,  ij.  58.  Phil, 


IN  VULGAR  FRACTIONS.  115 

48.  If  I  draw  a  bill  of  exchange  for  537!.  TOS  6d. 
Maflachufetts,  to  he  paid  in  Ireland, at  i2^Tlj\.  Maflachu- 
fetts, per  icol.  Irifh  ;  for  how  much  Inih  money  muit  I 
draw  the  bill  ?  Ans.  436!  141.  9J;d.  Irifii. 

49  Suppofe  a  bill  is  drawn  in  Ireland,  and  payable  in 
Bollon,  for  673!.  I2s.  6d.  Irifh  ;  how  much  MafTachu- 
fetts  money  comes  it  to,  the  exchange  at  8i^l  Irilh.  per 
lool.  MaiTachufetts  ?  Ans.  829!.  is.  6T6Td.  Muff. 

50.  If  I  give  D.I  75c.  for  3  yards,  how  many  yards 
may  I  have  for  D-i8o  ?  Ans  3c8yds.  2qrs.  i^n. 


RULE  OF  THREE  DIRECT  IN  VULGAR 
FRACTIONS. 

RULE* 

Having  made  the  neceflary  preparations,  as  directed  in 
Multiplication  and  Divifion  of  Vulgar  Fractions,  ftate  your 
queftion  as  in  whole  numbers,  and  invert  the  firft  term  of 
the  proportion  ;  then  multiply  the  three  terms  continually 
together,  and  the  product  will  be  the  anfwer. 

EXAMPLES. 

1.  If  4-  of  a  yard  cod  4  of  a  £.  what  will  T9r  of  an  Ell 
Englifh  co a  ? 

5X4X1 

$yd.=4  of  £  of  t~ =  i  ell  Eng. 

8X1X5 
E.  Eng.  £  E.  Eng.  2Xjx  9 

Asi:    4    ::    ^     :   MxA= cz^Wiyg.  ,Jd. 

1X7X15 
-§-  anfwer. 

2.  If  |yd.  cod  ^dol.  what  will  40^  yards  come  to  ? 

Ans.  D-59  6c.   2-Jm. 

3.  A  perfon  having  £  of  a  veffel,  fells  f  of  his  (hare 
for  D.I 080}  ;  what  is  the  whole  veflel  worth  ? 

Ans.  D.2026  250. 

4.  A  merchant  fold  5^-  pieces  of  cloth,  each  containing 
I2|yds.  at  12  Jc.  per  yard  ;  what  did  the  whole  amount 
to  ?  Ans.  D.8  82-£c. 

*  This  rule  and  the  next,  depend  upon  the  fame  principle  95  the 
E.ule  of  Three  w  \vhole  numbers, 


n<5  RULE  OF  THREE  DIRECT 

5.  A  buys  of  B  560*!.  bank   ftock,  at  85!!.  per  cent  j 
what  comes  it  to  ?  Ans.    480!.  7*   6^-d. 

6.  A  merchant    bought  a   number  of  bales  of  velvet, 
each  containing  « 29^-  yards,  at  the  rate  of  7  dollars  for  $ 
yards,  and  fold  them    out   at  the  rate   of  1 1  dollars  for  7 
yards ;  and  gained  200  dollars  by  the  bargain  ;  how  ma- 
ny bales  were  there  ? 

yds.       Dol.       yds.       Dol. 
As  7     :     if     ::     5     :     6£ 

Sold   5  yards  for         7!   dollars. 
Bought  5  yards  for     7     dollars. 

In  5  yards  gained         -y  dollar. 
Dol.      yds.         Dol.         yds. 
As  4     :     j>     ::     200     :      n66|,  and, 

yds.          B.       yds.        B. 
As   129*7  :  4-  ::   1 166-3-  :  9  anfwer. 

Although  the  method  before  laid  down  be  univerfally 
applicable,  yet  there  are  other  methods  more  ready  and 
expeditious  in  fome  particular  cafes. 

RULE    I. 

If  the  firft  and  third  terms  be  fractions,  and  the  fecond  a 
whole  number,  reduce  the  fir II  and  third  to  one  com  -non 
denominator,  then,  rejecting  the  denominators,  mike  the 
numerator  of  the  fink,  ihe  tirft.  term,  and  the  numerator 
of  the  third,  the  third  term,  and  work  as  in  whole  num- 
bers. 

If  4-  yard  cod  95.  what  coft  T'T  yard  at  that  rate  ? 
4— A4  and  yV—iJ*     Now,  As  15  :  93.  .:  14  :  8s.  4! d.  Ans. 


RULE    II. 

If  of  the  firft  and  third  terms,  one  be  I,  and  the  other 
a  fra&ton  ;  put  the  denominator  of  the  fradrm  instead 
of  i,  an  i  the  numerator  in  the  place  of  the  fraction,  and 
work  as  in  whole  numbers,  as  before. 

If  i  acre  of  land  coit  1 2!.  what  will  |-  of  an  acre  coft  at 
that  rate  ? 

Den.    £  Num.  £•  s. 
As  #  :   12  ;;  5  ;  7   10  Asa 


IN  DECIMALS.  M7 

RULE  III. 

If  the  fecond  term  be  a  fraction  likewife,  (that  is,  if  all 
the  terms  be  fradlions)  having  reduced  the  fir  ft  and  third 
to  one  common  denominator,  multiply   the  numerator  of 
the   firft   term   by  the  denominator  of  the  fecond,   for  a 
divifor  ;  and  the  numerator  of  the  third  by  the  numerator 
of  the  fecond,  for  a  dividend  ;  divide  the  laft  product  by 
the  firft,  and  the  quotient  will  be  the  anfwer. 
If  4r  yard  of  cloth  coft  £i  what  coft  -J-  yard  ? 
•j— J,  which  reduces  it  to  a  common  denominator  ;  then 
As  4     :    |     :;     7 
4  3 

16  16)21(1^5-1=^65,  3d.  Ans. 

16 


RULE  OF  THREE  DIRECT  IN  DECIMALS. 

RULE. 

Having  reduced  your  fractions  to  decimals,  and  ftated 
your  queftion  as  in  whole  numbers,  multiply  the  fecond 
and  third  together  \  divide  by  the  firft,  and  the  quotient 
will  be  the  anfwer. 

EXAMPLES. 

i.  If  |-  of  a  yard  coft  ^2  °f  a  pound  j  what  will  9! 
yards  come  to  ? 

1=5-625,  rW'583+,  and  |=-667— . 
As  '625   :  -583  ::  9-667 

•583 

29001 
7733^ 
48335 

•625)563586 1 (9-0-1 7-f=9l.  os  4d.  Ans, 
5625, 
1086 
625 
461 1 
4375 
236 


u8        RULE  OF  THREE  DIRECT,  &c. 

2.  If  ioz.  of  filver  coft  6?.  8d.   what  is  the  price  of  a 
bowl,  which  weighs  lib.   7oz.  I3gr.         Ans.  61.  6s.  icd. 

3.  If  9^-  yards  coft  D.I  £  25c.  what  will  1  yard   come 
to  ?  Ans.  57c.  6Yjni. 

4.  There  is  a  ciilern,  which  has  3  cocks,  the  firft  will 
emyty  it  in  £  of  an  hour,  the  fecond  in  -^,  awd  the  third 
in  li  hour-:  In  what  time  will  it  be  emptied  if  all  three 
run  together 

025          t     ::     i     :     4 
As   <  -75          i     ::     i     :     1*333-1- 
C.i'5          i     :;     i     :     o%607 — 

6  Cifterns 
Gift.     ho.      Gift. 

As  6  :  i  ::  i  :  -1667  —  =  lomin.  Ans. 
5',  A  conduit  has  a  cock,  which  running  into  a  cif- 
tern, will  fill  it  in  12  minutes  :  This  ciftern  has  three 
cocks  ;  the  firft  will  empty  it  in  i^  of  an  hour,  the  fec- 
ond in  37  j  minutes,  and  the  third  in  -J  an  hour  :  In  what 
time  will  the  ciftern  be  filled,  if  all  four  run  together  ? 

the  filling  cock. 
A     I  1-25  :    i   ::   i   :  0-81 

••'  'emptying  cocks. 


{•2  :  i  ::  i  :  5 

1*25  :  i  ::  i  :  o*8"l 

•625  :  i  ::  i  :  i  6  > 

•5  :  i  ::  i  :  2     J 


I 


4'4 

15     cifterns  filled  in  an  hour. 
4*4  ditto  emptied  in  do. 
•6     ciftern  difference. 
Gift.    h.  Gift. 

Then,  as  -6  :  i  ;V  i  :   1-67— =ih.  4om.  Ans. 
6.  If  19  yards  coft  D. 25*75,    what  will  435^  yards, 
come  to  ? 

yds.     D.  d.  c.        yds. 
As  19  :  25-  7  5   ::  435-5^ 

21775 


21775 
8710 
D.    d.  c.  m. 

2     I    7T2d 


RULE  OF  THREE  INVERSE.          1 19 

7.  y  345  yards  of  tape  coft  D-5'  id.    70.  5m.  what 
will  i  yd.  coft  ? 

yds.     D.  yds.     cm. 

As  345  :  5-175  ::   i   :  -015  Ans. 

8.  If  I  give  D.i2  8d.  2c.  5m.  for  675  tops,  how  ma- 
ny tops  will  19  mills  buy  ?  Ans.   i  top. 


RULE  OF  THREE  INVERSE, 

OR  RECIPROCAL  PROPORTION, 

TEACHES,  by  having  three  numbers  given  to  find  a 
fourth,  which  fhall  have  the  fame  proportion  to  the  fec- 
ond,  as  the  firfl  has  to  the  third. 

Therefore,  the  greater  the  third  term  is,  in  refpecl  to 
the  firft,  the  lefs  will  the  fourth  term  bes  in  refpecT:  to  the 
fecond  ;  or,  the  lefs  the  third  term  is  in  proportion  to 
the  firft,  the  greater  the  fourth  muft  be  in  proportion 
to  the  fecond  ;  and  this  is  called  reciprocal,  inverted^  or 
indirett  Proportion. 

The  principal  difficulty^  that  will  enibarrafs  the  learner 
will  be,  tt>  diftinguifh  when  the  proportion  is  direft,  and 
when  indirect.  This  is  done  by  an  attentive  confidera- 
tion  of  the  fenfe  and  tenour  of  the  queftion  proposed  :  For 
if  thereby  it  appears  that,  when  the  third  term  of  the 
dating  is  lefs  than  the  firft,  the  anfwer  muft  be  lefs  than 
the  fecond  ;  or,  when  the  third  is  greater  than  the  firft, 
the  anfwer  muft  be  greater  than  the  fecond  ;  then  the 
proportion  is  direct  :  But,  if  the  third  be  lefs  than  the 
firft,  and  yet  the  fenfe  of  the  queftion  requires  the  fourth 
to  be  greater  than  the  fecond  ;  or  the  third,  being  great- 
er than  the  firft,  the  anfwer  muft  be  lefs  than  the  fecond, 
the  proportion  is  inverfe. 

RULE.* 

State  and  reduce  the  terms  as  in  the  Rule  of  Three 
Direct  ;  then,  multiply  the  firfl  and  fecond  terms  together, 

*  Thereafou  of  this  rule  may  be  explained  from  the  principles  of 
Compound  Multiplication  and  Compound  Divifion,  in  the  fame 


120  RULE  OF  THREE  INVERSE. 

and  divide  the  product  by  the  third  ;  the  quotient^will  be 
the  anfwsr  in  the  fame  denomination  as  the  middle  term 
was  reduced  into. 

If  there  be  fractions  in  your  queftion,  they  muft  be 
Stated  as  before  directed,  and  if  they  b$  vulgar,  invert  the 
third  term  :  Then  multiply  the  three  terms  continually 
together,  and  the  product  will  be  the  anfwer. 


EXAMPLES. 

i.  How  much   ftialloon,   that  is  |  yard  wide,  will  line 
6;|  yards  of  cloth  which  is  14-  yard  wide  ? 

yd.  yds.  qrs.  qrs.  qrs;  qrs. 

As    ij  :  6|  ::  3  As  5   :   27  ::  3 

44  5 

S      27  3)'3S 

4)45 

1  1  £  yards,  Ans. 

The  fame   by  Vulgar  Fractions.     « 

Firft,  i  Jc^,  6|=V,  and  3qrs.=J,     Then, 

5X27X4  yds, 


A_     S      .     27      ..     3 

As  Tf  :    ^    ::  ^. 

4x4x3 

the  anfwer, 


manner  as  the   diredl  rule — -For  example.  If  4  men  can  do  a   piece 
of  work  in  ia  days,  in  what  time  will  8  men  do  it  ? 

4X1* 

As  4  men  :  ii  days  ::  8  men  : =6  days,  the  Anfwer. 

8 

And  here  the  product  of  the  firft  and  fecond  terms,  that  is,  4  times 
i  a,  or  48,  is  evidently  the  time  in  which  one  man  would  perform 
the  work.  Ther efore^  8  men  will  do  it  in  one  eighth  part  of  th* 
time,  or  6  days. 


RtfLE  OF  THREE  INVERSE. 

The  fame  by  Decimal  Fractions. 
Ij=r25,  6J-6-75,  and  3qrs.='75.     Then 
As    1-25  :  6-75  ::  -75 


75)8-4375(1 1 '25  yard',  Anfwer, 


93 

75 

187 
150 

375 
375 

2.  What  length  of  board,  7  inches  wide,  will  make  a 
fqaare  foot  ? 

In.br.  in.len.     in.br.    in.len. 
As  12     :     12     ::     7*     :     19^  Ans. 

3.  How  many  yards  of  carpet,  2|  feet  wide  will  cover 
a  floor  which  is  1 8  feet  long  and  1 6  feet  wide  ? 

ft.         ft.         ft.  ft.         yds. 
As    1 6    :    18    ::    2|X3    :    34^  Ans. 
NOTE,    I  multiply  2^by  3,  becaufe  3  feet=i  yard. 

4.  Suppofe    I    lend   a  friend  .£350  for  5  months,  he 
promifing   the   like  kindnefs  ;  but,  when   requeued,  can 
fpare  but  £125;  how  long  may  I  keep  it  to  balance  the 
favour  ?  £        Mo.          £          Mo. 

As  350    :    5    ::     125    :    14  Ans. 

5    Suppofe  450  men  are  in  a  garrifon,  and  their  pro- 

vifions  are  calculated  to  laft  but  5   months  ;  how   many 

muft  leave  the  garrifon,  that  the  fame  provisions  may  be 

fufficient  for  thofe  who  remain  9  months  ? 

Mo.   M.    Mo.    M.  M. 

As  5  :  450  ::  9  :  250,  and  450—250=200  men  An?, 
L 


122  RULE  OF  THREE  INVERSE. 

6.  If  a  man  perform  a  journey   in  15  days,  when  the 
day  is  1  2   hours  long,   in  how   many  days  will  he  do  it, 
when  the  day  is  but  to  hours  ?  Ans.  1  8  days. 

7.  If  a  piece  of  land,  40  rods  in  length,  and  4  in  breadth, 
make  an  acre,  how  wide  muft  it  be,  when  it  is  but  19  rods 
long,  to  make  an  acre  ?  Ans.  8  rods  6ft.  n^m. 

8.  If  when  wheat  is  D.  i  per  bulhel,  the  two  penny  loaf 
weigh  9  6oz.  what  ought  it  to  weigh,  when  wheat  is  D.I 
2JC.  perbufhel?  Ans.  yoz.  i3pwt.   i4*4gr. 

9.  If  a   piece   of  board  be  30   inches  in   length,  what 
breadth  will  make  i^fquare  foot  ?  Ans.  7-2  inches. 

10.  If  9  men  can  build  a  houfe  in  5  months,  by  work- 
ing 14  hours  per  day,  in  what  time  will  the  fame  number 
of  men  do  it,  when  they  work  only  10  hours  per  day  ? 

Ans.  7  months. 

n.  A  wall,  which  was  to  be  built  24  feet  high,  was 
raifed  8  feet  by  6  men,  in  1  2  days  :  How  many  men  muft 
be  employed  to  finifh  the  wall  in  four  days  ? 

ft.     m.       ft.  _         m. 
As  8   :   6   ::    24  —  8   :    12  to  finifh  it  in  1-2  days  4  and 

d.         m.        d.        m. 
As   12     :     12     ".4    :    36  to  finifli  in  4  days. 

12.  There  is  a  ciftern  having  a  pipe,  which  will  empty 
it  in  6  hours  :  How  many  pipes  of  the  fame  capacity,  will 
empty  it  in  20  minutes  ? 

h.        pi.          mi.         pi. 
As  6     :      i     ::     20     :     18  Ans. 

13.  What  number  of  men  muft  be  employed  to  finifh 
in  9  days  what  15  men  would  be  30  days  about  ? 

Ans  50  men. 

14.  If  a  field  will  fetd  6  cows  91  days,  how  long  will 
it  feed  21  cows  ?  Ans.  26  days. 

15.  How  much  in  length,  that  is  8  5-  inches  broad,  will 
make  a  foot  fquare  ?  Ans.  i6|f  inches. 

1  6.  How  much  in  length  that  is  13  J  poles  in  breadth, 
will  make  a  fquare  acre  ?  Ans. 


17.  A  regiment  of  foldiers,  confiding  of  745  men,  is  to 
be  clothed,  each  fuit    to  contain  3^  yards  or  cloth,  which 


DOUBLE  RULE  OF  THREE.  123 

is  i  g  yard  wide,  and  lined  with  fhalloon  J  yard  wide  ;  how 
many  yards  of  fhalloon  will  line  them  ? 

As  745X3^   :    i|    ::   -J-   :    409 7 J  yards,  Ans. 

1 8.  If  a  fuit  of  clothes  can  be  be  made  of  4^  yards  of 
cloth,  1 1  yard  wide  ;  how  many  yards  of  coating  £  of  a 
yard  wide,  will  it  require  for  the  fame  perfon  ? 

Ans.  6yds«   iqr.  3^. 


COMPOUND    PROPORTION, 

0*  DOUBLE  RULE  OF  THREE, 

TEACHES  to  refolve  fuch  questions  as  require  two,  or 
more,  ftatings  by  fimple  proportion  ;  and  that,  whether 
direft  or  inverfe  :  It  is  compofed  (commonly)  of  5  num- 
bers to  find  a  fixth,  which  if  the  proportion  be  direct,  mud 
bear  fuch  proportion  to  the  4th  and  5th  as  the  $d  bears 
to  the  lit  and  zd  ;  but  if  inverfe,  the  6th  number  muft  bear 
fuch  proportion  to  the  4th  and  jth,  as  the  firft  bears  to 
the  zd  and  3d. 

RULE. 

Always  place  the  three  conditional  terms  in  this  order  : 
That  number,  which  is  the  principal  caufe  of  gain,  lofs  or 
aclion,  ponenes  the  firft  place  ;  that,  which  denotes  the 
fpace  of  time,  diftance  of  place,  rate,  medium  or  mean  of 
a&ion,  the  fecond  ;  and  that,  which  is  the  gain,  lofs  or 
adion,  the  third  :  This  being  done,  place  the  other  two 
terms  which  move  the  queftion,  under  thofe  of  the  fame 
name,  and  if  the  blank  place,  or  term  fought,  fall  under 
the  third  place,  then  the  question  is  in  direct  proportion  : 
therefore, 

Rule  i. — Multiply  die  three  laft  terms  together,  for  a 
dividend,  and  the  two  firll  for  a  divifor  : — But  if  the 
blank  fall  under  the  firft  or  fecond  place  ;  then,  the  pro- 
portion is  inverfe  ;  therefore, 

Rule  2 — Multiply  the  firft,  fecond  and  laft  terms  to- 
gether for  a  dividend,  and  the  other  two  for  a  divifor,  and 
the  quotient  will  be  the  anfwer. 


124  DOUBLE  RULE  OF  THREE. 

EXAMPLES. 

i.  If  D.  100  gain  D.6  in  a  year  ;  what  will  D  400  gain 
:n  9  months  ? 

D.P.   Mo.  D.  Int. 

f    ("Terms   in  the  fuppofition,  or  condi* 
ioo  :   it::  6    J     tional  terms. 

400  :     9  Terms  which  move  the  queftion. 

Here,  the  blank  falling  under  the  third  place,  the  quef- 
tion is  in  direct  proportion,  and  the  anfwer  muii  be  found 
by  the  firft  Rule  ;  therefore, 

4ocx  9x6=21600  For  the  dividend,   and, 
100X12     =1200  For  the  divifor. 

See  the  work  at  large, 

D.  Pr.  Mo.    D.  Int, 
ioo    ;    12  ::  6 

400    :     9 
9 

ioo    3600 

12  6 

i2!co)2i6|oc(t8D.  Ans, 

J2 


96 


^.  If"  D.IOO  will  gain  D.6  in  a  year  ;  in  what  time 
0.400  gain  D.i8  ? 

D.      Mo.     D. 

ioo  :  12  ::     6   Terms  in  the  fuppofition. 
400  :         ::   18    Terms  which  move  the  queftion. 
Here  the  blank  falling  under  the  2d  place,  the  queftion 
is  in  reciprocal  or  inverfe  Proportion,  and  the  anfwer  mult 
be  fought  by  the  fecond  rule  :  therefore, 

100X12X18=21600  For  the  dividend. 
40OX  6      =  2400  For  the  divifor. 


DOUBLE  RULE  OF  THREE.  125 


D.Pr.   Mo. 

D.Int. 

100  :  12 

::  6 

400  : 

::  18 

6 

12- 

400 

216 

100 

24|oo)2i6|oc(9  months,  Ans. 
216 

3.  What  principal,  at  6  per  cent,  per  ann.  will  gain 
D.I 8  in  9  months  i 

Pr.         Mo.         Int, 
100     :     12     ::     6 

9     ::    18 

12 

9    216 
6    100 
• D. 


54)21600(400  Anj. 
216 

Here  the  blank  falling  under  the  nrft  place,  the  propor- 
tion is  inverfe,  and  the  anfwer  found  by  the  fecond  rule, 
as  in  the  lait  example. 

4.  If  0.400  gain  D.I 8  in  9  months ;  what   is  ths  rate 
per  cent,  per  annum  ? 

Pr.  Mo.  Int. 
400  :  9  ::  18 
100  :  12  ::  D  6  Ans. 

5.  If  8  men  fpend  £$2  in  13  weeks  ;  what  will  24  men 
fpend  in  52  weeks  ?  Ans.    2*3  84- 

6.  If  the   freight   of  9hhds.  of  fugar,  each   weighing 
I2cwt.  20  leagues,  coll  D-5O  ;  what  muft  be  paid  for  the 
freight  of  50  tierces  ditto,  each  weighing    2icwt.    100 
leagues  ?  Ans.  0,189  350., 

L    2 


126  CONJOINED  PROPORTION. 

7.  If  6  men  build  a  wall  20  feet  long,  6  feet  high   and 
4  feet  thick,  in  16  days,  in  what  time  will   24  men   build 
one  200  feet  long,  8  feet  high,  and  6  feet  thick  ? 
m.     da.        ft. 
6  :   1 6  ::      20X6X4 
24  :         ::    200X8x6  80  days,  Ans. 

COMPARISON  OF  WEIGHTS  AND   MEASURES. 

EXAMPLES. 

i.  If  78  pence  MafTachufetts  be  worth  one  i  French 
crown,  how  many  Maflaehufetts  pence  are  worth  320 
French  crowns  ? 

F.  cr.     d.         F.  cr. 

As    i     :     78    ::    320 

78 


24960    Ans, 

1.  If  24  yards  at  Bofton  make  16  ells  at    Paris,  how 
many  ells  at  Paris  will  make  128  yards  at  Bofton  ? 

Boft.         Par.  Boft.  Par. 

As   24yds.  :    i6ells   ::    1 28yds    :    Shells,  Ans. 

2.  If  6clb.  at  Bofton  make  j61b.  at  Amfterdam,  how 
many  Ib.  at  Bofton  will  be  equal  to  350  at  Amfterdam  ?. 

Ans.  375lb.  Bofton. 

4.  If  gjib.  Flemifli  make  loolb.  American,  how  many 
American  Ibs.  are  equal  to  fjolb.  Flemiih  ? 

Ans.  578j-|-lb.  American. 


CONJOINED  PROPORTION 

IS  when  the  coins,  weights  or  meafures  of  feveral  coun- 
tries are  compared  in  the  fame  question  ;  or,  in  other 
words,  it  is  joining  many  proportions  together,  and  by 
the  relation,  which  feveral  antecedents  have  to  their  con« 


CONJOINED  PROPORTION.  127 

fequents,  the  proportion  between  the  firft  antecedent  and 
the  laft  confequent  is  difcovered,  as  well  as  the  proportion 
between  the  others  in  their  feveral  refpects. 

This  rule  may  generally  be  fo  abridged  \ty  cancel* ing 
equal  quantities  on  both  fides,  and  abbreviating  commen- 
furables,  that  the  whole  operation  may  be  performed  with 
very  little  trouble,  and  it  may  be  proved  by  as  many 
ftatings  in  the  Single  Rule  of  Three,  as  the  nature  of  the 
queftion  may  require. 

CASE    I. 

When  it  is  required  to  find  how  many  of  the  fir  ft  fort  of 
coin,  weight  or  meafure,  mentioned  in  the  queftion,  are 
equal  to  a  given  quantity  of  the  laft. 

RULE. 

Place  the  numbers  alternately,  that  is,  the  antecedents 
at  the  left  hand,  and  the  confequents  at  the  right,  and  let 
the  laft  number  ftand  on  the  left  hand  ;  then  multiply  the 
left  hand  column  continually  for  a  dividend,  and  the  right 
hand  for  a  divifor,  and  the  quotient  will  be  the  anfwer. 

EXAMPLES. 

i.  Suppofe  100  yards  of  America^ioo  yards  of  Eng- 
land, and  100  yards  of  Eugland=5o  canes  of  Thouloufe, 
and  100  canes  of  Thouloufe=i6o  ells  of  Geneva,  and  ico 
ells  of  Genevan  200  ells  of  Hamburgh  :  How  many 
yards  of  America  are  equal  to  379  eils  of  Hamburgh  ? 
Antecedents.  Confequents.  Abridged. 

100  of  America     =  ico  of  England.  Ant.  Con. 

TOO  of  England     =     50  of  Thouloufe.  5         8 

i oo  of  Thouloufe  =  160  of  Geneva.  379 

100  of  Geneva      =  200  of  Hamburgh. 
379  of  Hamburgh  ? 

379X5 

Therefore, =236^5.   of  America=379  ells  of 

8 
Hamburgh. 

ILLUSTRATION. 

The  two  i  cos  of  bath  Tides  cancel  each -other.  Let  the 
laft  cyphers  of  the  three  next  antecedents  and  confequents 
be  cancelled,  which  is  dividing  by  10.  Thea  divide  the 


128  CONJOINED  PROPORTION. 

fecond  antecedent  and  confequent  by  5,  and  the  quotients 
•will  be  ^  on  the  fide  of  the  antecedents,  and  i  on  the  fide 
of  the  confequents  ;  then  2  will  meafure  the  third  antece- 
dent and  confequent,  and  the  quotients  will  be  5  and  8. 
10  will  meafure  the  4th  antecedent  and  confequent,  and 
the  quotients  will  be  i  and  2.  Now,  there  being  two  left 
on  each  fide,  they  cancel  each  other,  and  as  there  is  na 
farther  room  for  abridging  by  reafon  of  that  odd  number 
379,  the  operation  is  finiflied,  and  the  anfwer  found  as 
before. 

2.  If  i2lb.  at  Bofton  make  lolb.  at  Amfterdam,   and 
soolb.  at  Amfterdam  i2olb.  at  Paris  :  How  many  Ib.  at 
Bofton  are  equal  to  8olb.  at  Paris  ?  Ans.  8olb. 

3.  If  4olb.  at  Newburyport   make  36  at  Amfterdam, 
and  Qolb.  at  Amfterdam  make    n6at  Dantzick  :  How 
many  Ib.  at  Newburyport  are  equal  to  26olb.  at  Dantzick? 

Ans.  224^, 

CASE  II. 

When  it  is  required  to  find  how  many  of  the  laft  fort 
of  coin,  weight  or  meafure,  mentioned  in  the  queftion,  are 
equal  to  a  given  quantity  of  the  firft. 

RULE. 

Place  the  numbers  alternately,  begining  at  the  left  hand, 
and  let  the  laft  number  ftand  on  the  right  hand  ;  then 
multiply  the  firft  row  for  a  divifor,  and  the  fecond  for  a 
dividend. 

EXAMPLES. 

i.  Suppofe  ico  yards  of  America=ioo  yards  of  Eng- 
land, and  100  yards  of  Englana=5o  canes  of  Thouloufe, 
and  100  canes  of  Thouloufe=i6o  ells  of  Geneva,  and 
joo  ells  of  Geneva=2Oo  ells  of  Hamburgh  :  How  many 
ells  of  Hamburgh  are  equal  to  2363-  yards  of  America  % 

Ante.  Confe.  Abridged^. 

100  Amer.  =100     Eng.  Ant.     Con. 

ico  Eng.     =50     Thoul.  5 

j  oo  Thoul.  =j  60     Gen. 
100  Gen.     =200     Ham.  236^X8 

Amer. 


CONJOINED  PROPORTION.   .        129 

Th'is  needs  no  further  illuftration.  The  learner  will 
readily  fee  that,  this  cafe  being  the  reverfe  of  the  former, 
they  are  proofs  to  each  other. 

2.  If  tzlb.  at   Bofton  make  lolb.  at  Amfterdam,  and 
loolb.  at  Amfterdam    I2olb.   at  Paris:    How  many  at 
Paris  are  equal  to  8olb.  at  Bofton  ?  Ans.  8olb. 

3.  If  4olb.  at  Newburyport    make  36  at  Amfterdam, 
and  9olb.  at     Amfterdam  make  1 16  at  Dantzick  :  How 
many  Ib.  at  Dantzick  are  equal  to   244  at  Newburyport  ? 


ARBITRATION  OF  EXCHANGES. 

By  this  term  is  underftood  how  to  choofe,  or  determine 
the  bed  way  of  remitting  money  abroad  with  advantage  : 
which  is  performed  by  conjoined  proportion  :  Thus, 

Suppofe  a  merchant  has  effects  at  Amfterdam  to  the  a- 
mount  of  3530   dollars,  which  he  can  remit  by  way  of 
Lifbon  at   840   rees  per  dollar,  and  thence  to  Bofton,  at 
8s.  rd.  per  milree  (or  1000  rees)  :   Or,  by  way  of  Nantz, 
at  5f   livres  per  dollar,  and  thence  to  Bofton  at  6s.   8d. 
per  crown  ;    it  is  required  to  arbitrate  thefe  exchanges3 
that  K,  to  choofe  that  which  is  moft  advantageous  ? 
i  dollar  at  Amfterdam  =840  rees  at  Lifbon. 
i  coo  rees  at  Lifbon      =  pyd.  at  Bofton. 

3530  dollars  at  Amfterdam, 
840X97X3530 
=£1198  8s.  8/o-d.  by  way  of  Lifbon. 

IOOOXI 

i  dollar  at  Amfterdam  =5!  livres  at  Nantz. 
6  livres  at  Nantz  =^8o  pence  at  Bofton. 

3530  dollars  at  Amfterdam. 

=,£1059  by  way  of  Nantz. 
1X6 

Here  it  may  be  obferved  that  the  difference  is  £139  8s. 
8/^d.  in  favour  of  remitting  by  way  of  Lifbon  rather 
than  by  Nantz,  which  depends  on  the  courfe  of  exchange, 
at  that  time  ;  but  the  courfe  may  vary  fo,  that,  in  a  fhort 


130  SINGLE  FELLOWSHIP. 

time  by  way  of  Nantz,  may  be  better  ;  hence  appears  the 
neceffity  and  advantage  of  an  extenfive  correspondence,-, 
to  acquire  a  thorough  knowledge  in  the  courfes  of  ex- 
change, to  make  this  kind  of  remittance, 


FELLOWSHIP. 

THE  Rules  of  Fellowfhip  are  thofe  by  which  the  ac- 
compts  of  feveral  merchants  or  other  perfons,  trading  in 
partnership,  are  fo  adjufted,  that  each  may  have  his  fhare 
of  the  gain,  orfuftain  his  fhare  of  the  lofs,  in  proportion 
to  his  fhare  of  the  joint  ftock,  together  with  the  time 
of  its  continuance  in  trade, 

SINGLE  FELLOWSHIP 

Is,  when  the  ftocks  are  employed  for  any  certain  equal 
time. 

RULE.* 

As  the  whole  ftock  is  to  the  whole  gain  or  lofs,  fo  is 
each  man's  particular  flock  to  his  particular  fhare  of  the 
gain  or  lofs. 

PROOF.  Add  all  the  particular  fhares  of  the  gain  or 
lofs  together,  and,  if  it  be  right,  the  fum  will  be  equal  to 
the  whole  gain  or  lofs. 

EXAMPLES. 

i.  Divide  the  number  360  into  four  fuch  parts,  which 
lhall  be  to  each  other,  as  3,  4,  5  and  6. 

As  3+4+5+6  :  360  ::    \M       HAnfwer. 


!3  :  6o-l 

4  :  80  / 

5   :  100  f 

6  :  i2oj 


360  Proof. 

*  That  their  gain  or  lofs,  in  this  Rule,  is  in  proportion  to  their 
ftocks,  is  evident :  For,  as  the  times,  in  which  the  ftocks  are  in 
trade,  are  equal, if  I  put  in  one  half  of  the  whole  ftock,  I  ought  to 
have  one  half  of  the  gain ;  if  ray  part  of  the  ftock  be  one  fourths 
my  ihare  of  the  gain  or  lofs  ought  to  he  one  fourth  alfo.  And  gen- 
erally the  fame  ratio  that  the  whole  ftock  has  to  the  whole  gain  or 
lofs,  muftcach  perfon's  particular  ftock  have  to  his  refpeftive  gain 
or  lofs. 


SINGLE  FELLOWSHIP.  131 

2.  B,  C?  O,  and  E  companied  ;  B  put  in  145!.  ;  C, 
219!.  ;  D.  378!.  and  £,-417!.  with  which  they  gained 
c6ol.  :  What  was  the  (hare  of  each  ? 

£     s.  d.      ftarcs. 

Whole  flock. 
As  145+2 19+378+417: 


569  —  Proof. 

3.  K,  L,  M,  and  N  are  concerned  in   a  joint  ftook  of 
D.iooo  ;  of  which  K's  part  is  D.  150;  L's  0.250  ;    M's 
0.275,  and  N's  325  ;  upon  the  adjuftment   of  their  ac- 
compts,  they  have  loft  0.337  500.     What  is  the  lofs  of 
each  ?  D.    c. 

Ans.  KO.5o62Jc.;  L  0.84  37^.  ;  M  0.92  8i£c. ; 
N  0.109  68*c. 

4.  F,  G  and  H  freighted  a  (hip  with  68900  feet  of 
boards  :     F   put  in    16520  feet;  G  28750;  and  H  the 
reft  }  but  in  a  ftorm  the  captain  threw  overboard  26450 
feet :  How  much  muft  each  fuftain  of  the  lofs  ? 

Ans.  F  6341 1  feet;  G  1 1036^,  and  H  9071^  do. 

5.  A  (hip,  worth  0.3000,  being  loft  at  fea,  of  which 
£  belonged  to  O,  ±  to  P,  and  the  reft  to  Q_:    What  lofs 
will  each  fuftain,  fuppofmg   0.450  to  have  been  infured 
iipon  her  ?  Ans.  O's  lofs  0.3 1 2  jcc. 

P's  937  50 

Q>  625 

6.  X  and  Y  venturing  equal  fums  of  money,  cleared 
by  joint  trade  O  140  ;  by  agreement,  as  X  did  the  bull- 
nefs,  he  was  to  have  8  per  cent,  and  Y  was  to  have  5  pa- 
cent.  :  What  was  X  allowed  for  his  trouble  ? 

O.  O.  D.        O. 

As  84-5     :     14-0     ::     8     :     86T7T  ;  and, 
As  8+5     :     140     ::     5     :     53!^ 

Ans.  0.32  300.  7-TTm« 

7.  A  bankrupt  is  indebted  to  R  £120,  to  S  £230,  to 
T  £ 340,  and  to  U  £490,  and  his  whole  eftate  amounts 
only  to  ^"560  :  How  muft  it  be  divided  among  his  cred- 
itors ? 

Ans.  R£58   18   u£  ;  S£ii2   19  7$;  T^*i67  04? 

U£22I     I    0$. 


1.3*  DOUBLE  FELLOWSHIP. 

8.  X,  Y,  and  Z  put  their  money  into  a  join   ftcck  5 
X  put  in  0.40  ;  Y  and  Z  together,  D.I7O  ;  they  gained 
D.I 26,  of  which  Y  took  0.42  :  What  did  X  and  Z  gain, 
and  Y  and  Z  put  in  refpeclively  ? 

As  D. 210  the  whole  ftock  :  D.I26  the  whole  gain  :: 
D.40  X's  (lock  :  D.24  X's  gain. 

As  D.24  X's  gain  :  D.4o  X's  flock  ::  D.42  Y's 
gain  :  D.7o  Y's  ilock.  Then  D.i7o— D.7o=D.ioo  Z's 
ftock  ;  and  whole  gain  D.I26— D.66  X's  and  Y's  gain= 
D.6o  Z's  gain. 

9.  E,  F,  G,  and  H  companled  ;    and  gained  a  Turn  of 
money,  of  which  E,  F,  and  G  took  £120  j  F,  G,  and  H, 
£180  ;  G,  H,  and  E,  £160  ;  and  H,  E,  and  F,  £140  ; 
what  diftinft  gain  had  each  ? 

The  fum  of  thefe  4  numbers  is  £600,  and  as  each 
man's  money  is  named  3  times,  therefore  |,  viz.  £200,  is 
the  whole  gain.  Therefore  £100 — £120  E's,  F's,  and 
G's  gainrr£8o  H's  gain  ;  £200— £180  F's,  G's,  and  H's 
gain=£2o  E's  gain  ;  £200 — £160  G's,  H's  and  E's  gain 
=£40  F's  gain;  and  £200— £140  H's,  E's  and  F's  gain 
?=£6o  G's  gain. 


DOUBLE  FELLOWSHIP* 

Or  Fellowfoip  'with  Time,  is  occafionedby  the  fhares  of 
partners  being  continued  unequal  times. 

RULE. 

Multiply  each  man's  ftock,  or  fiiare,  by  the  time  it  was 
continued  in  trade.  Then, 

As  the  whole  fum  of  the  products,  is  to  the  whole  gain 
or  lofs,  fo  is  each  man's  parti cular  product,  to  his  particu- 
lar fhare  of  the  gain  or  lofs. 

EXAMPLES. 

T.  A,  B,  and  C  hold  a  paflure  in  common,  for  which 
they  pay  40!.  per  annum  ;  A  put  in  9  oxen  for  5  weeks  ; 

*  When  times  are  equal,  the  fiiares  of  the  gain  or  lofs  are  evi- 
dently as  the  ftocks,  as  in  Single  Fellowship ;  and  when  the  flocks 
are  equal,  th  v.ares  are  as  the  times  ;  wherefore,  when  neither  air 
equal,  the  fiiares  muft  be  as  their  produces. 


DOUBLE  FELLOWSHIP 


B,  12  oxen  for  7  weeks  ;  and  C,  8  oxen  for  16  weeks. 
What  muft  each  pay  of  the  rent  ? 

9X5=45.     12=7=84,  and  8X16=128  ;  then 

128+84+45=257. 
As  257  :  40  ::  45  As  257  :  40::  84  As  257  :  40  ::  128 


45 

200 
160 

257)1800(7 
1709 

i 

20 

257)20(0 
12 

257)240(0 

4 

257)9^(3 
771 

189 


84 

1  60 


257)3360(13 
257 

790 

771 

19 

20 

257)380(1 
257 

123 

12 


1285 

191 

4 

257)764(2 
5*4 


40 

257)5120(19 
257 

2550 


237 

20 


257 

2170 
2056 


12 


1285 
~ 

4 

257)332(1 
257 


250  75 

2.  Four  merchants  traded  in  company  ;  L  put  in  400 
dollars  for  5  months,  M,  D.6oo  for  7  months,  N,  0.960 
for  8  months,  and  O  D.I2OO  for  9  months  ;  but  by  mis- 
fortunes at  fea,  they  loft  0.750  :  What  muft  each  fuftain 
of  the  lofs  ? 

•94  93C<  644m.    N  0.227  840. 
142  40      5TV        Q       284  8 1 
M 


A       CL  D. 

Ans'lM    1 


o|| 


?34         FELLOWSHIP  BY  DECIMALS. 

3.  A,  with  a  capital  of  lool.  began  trade  January  ift, 
1787,  and  meeting  with  fuccefs  in  his  bufinefs,  he  took  in 
B  as  a  partner  on  the  firft  day  of  March  following,  with 
a  capital  of  150!.     Three  months  after  that,  they  admit 
C  as  a  third  partner,  who  brought  into  flock   i8ol.   and 
after  trading  together  until   the  firft  of  January,  1788, 
they  found  there  had  been  gained  fmce  A's  commencing 
bufinefs,  177!.  133.     How  muft  this  be  divided  among 
the  partners?  fA  53!.    i6s.     8d. 

Ans.  <  B   67       5      10 
1C    56     10        6 

4.  Two  merchants   entered    into    partnerfhip   for   18 
months  ;  K  firft  put  into  ftock  0.400,  and  at  the  end  of 
8  months  he  put  in  200  more  ;  L,  at  firft  put  in  D.i  100, 
and  at  the  end  of  4   months   took  out   D.28o.     Now  at 
the  expiration  of  the   time,    they  found  the/  had  gained 
I).  1052  :  What  is  eacli  man's  juft  fhare  ? 

Ans.  K,  0.385  poc.  L,  D.666   ice. 

5.  N  and  O  companied  ;  N  put  in,  the  ift  of  January, 
150!.  but  O  could  not  put  in  any  till  the    ift  of  May  : 
What  did  he  then  put  in,  to  have  an  equal  fhare  with  N 
at  the  jear's  end  ? 

M.     £       M.     150X12 

As   12  :   150  ::  8  :  =£^25  Ans. 

8 

FELLOWSHIP  BT  DECIMALS. 

RULE. 

Divide  the  whole  gain,  or  lofs,  by  the  whole  ftock,  or 
fum  of  all  the  products,  as  the  cafe  requires,  and  the 
quotient  multiplied  feverally  by  each  man's  ftock,  or  pro- 
duct, will  give  the  gain  or  lofs  of  each. 

EXAMPLES. 

i.  H,  I,  and  K  companied  ;    H  put  in  40!.  55.  ;  T, 
Sol.  i os.  ;  and  K,  i6il.  with  which  they  gained  120!.  : 
What  is  each  man's  fhare  of  the  gain  ? 
H's  ftock  ss    40*25 
I's     do.      =:     80-5 
K's   do.     =  161" 

Sum  total  «=  28r75.)i2Ofooccoo('4259-f 


FELLOWSHIP  BY  DECIMALS.          135 

•4259 
80-5 

21295 
34272 

£34-28495 
£17-142475  20  £68-5699 

20 2O 


5-69900 


2-849500  12 

12 


1O-I94000 

4 

1-552 

6-776000 

Proof.     H's  gain  17!.  2s.  iod.-t-IJs  gain  34!   j 
K's  gain  681.  us.  4-Jd.=  ii9l.  193.  i  id. 

2.  L,  M,  and  N  companied  ;  L  put  in  0.400  for  8 
months ;  M,  0.300  for  9  months,  and  N,  1) .175  for  12 
months  ;  with  which  they  gained  0.720.  Required  ths 
Hi  are  of  each  ? 

D.  Mo.     Prod. 
L,     4ocx  8  =  3200  • 
M    3ocX  9  =  2700 
N     175X12  ~  2100 


Sum  of  Produfts  =  8ooo)72O'(-O9=quotient. 

3200  X  -09  =  288  =  L's  (hare,     "1 
2700  X  -09  =  243  =  M's  lhare,    J- Ans. 
2 1  OQ  X  -09  =   189  =  N's  Iharc^    j 

r  3.  An  infolvent  eftate,  amounting  to  0.633  6oc.  is  in- 
debted to  A  0.312  750.  to  B  O  297,  to  C  0.50  250.  to 
O  250.  to  E  0.200,  to  F  0142  500.  and  to  G  O.2  i  250.$ 
what  proportion  will  each  creditor  receive.' 

633-6 

" =;*6i3;5;  or, 

3 1 2-75-V297+50-25+  254-200+142-54-2 1-25 
6 ic.  8-|m.  on  a  dollar.     And, 


I36 


D.  c. 

312  75 1 
297- 
50-25 

•25 

200* 
142-50 
21-25 


PRACTICE. 

D.  c.    m. 

'Jy3'5r  4rV 

=  A's  proporfion 

183-76  S| 

=  B's  do. 

31*09    2r3,~ 

=  C's  do. 

c=.  <j        -15  4-J-V 

=  D's  do. 

1  12375 

=  E's  do. 

88-17   4 

=  Fs  do. 

T  ...    .      07 

^_     J  3   *  4       T"6~ 

=  G's  do. 

Proof  D. 63 3- 60 


PRACTICE 

IS  a  contraction  of  the  Rule  of  Three  Direct,  when 
the  firft  term  happens  to  be  an  unit,  or  i  ;  and  has  its 
name  from  its  daily  ufe  among  merchants  and  tradef- 
men,  being  an  eafy  and  concife  method  of  working  moft 
queftions,  which  occur  in  trade  and  bufinefs. 

The  method  of  proof  is  by  the  Rule  of  Three,  Com- 
pound Multiplication,  or  by  varying  the  order  of  them. 
GENERAL  RULE. 

1.  Suppofe  the  price  of  the  given  quantity  to  be  il.  or 
is.  &c.  then  will  the  quantity  itfelf  be  the  anfwer  at  the 
fuppofed  price. 

2.  Divide  the  given  price  into   aliquot  parts,  either  of 
the  fuppofed  price,  or  of  one  another,  and  the  fum  of  the 
quotients,   belonging  to  each,  will  be  the  true  anfwer  re- 
quired. 

EXAMPLE. 

What  is  the  value  of  468  yards,  at  2s.  9^d.  per  yard  ? 
£468  s.     d.         Anfwer  at  £  i   s.  d. 


zs.  6d.  is  I    =58   10  o  ditto  at  o  2  6 

3d.  is  Tfo  =     5170  ditto  at  o  o  3 

•£d.  is  YF  =     o     99  ditto  at  o  o  o^- 

The  full  price  =£64  16  9  °  2  9i 

In  this  example  it  is  plain,  that  the  quantity  468  is  the 
anfwer  at  il.  ;  confequently,  as  2s.  6d.  is  -J-  of  a  pound, 
•J  part  of  that  quantity,  or  58!,  IQS.  is  the  price  at  2s,  6d,  : 


PRACTICE. 


'37 


in  like  manner,  as  3d.  is  the  TV  part  of  2s.  6d.  fo  TV  part 
of  58!.  i  os.  or  5!.  175.  is  the  anfwer  at  3d.  ;  and  as  iqr. 
is  T^  of  3d.  fo  TT  of  5!.  175.  or  95.  9d.  is  the  anfwer  at 
jqr.  Now  as  the  fum  of  all  thefe  parts  is  equal  to  the 
whole  price  (25.  9id.)  fo  .the  fun*  of  .the  anivvers  belong- 
ing to  each  price  will  be  the  anfwer  at  the  full  price  .re- 
quired, and  the  fame  will  be  tru-  in  any  example  what- 
ever. 

GENERAL    RULE, 

To  find  the  value  of  goods  in  Federal  Money. — Mul- 
tiply the  price  and  quantity  together  ;  point  off  in  the 
product  for  denominations  lower  than  dollars,  as  many 
places  as  there  are  in  the  given  price  ;  or,  if  there  be  dec- 
imal places  in  the  quantity,  (or  lower  denominations  pre- 
vioufly  reduced  to  decimals)  according  to  multiplication 
of  decimals. 

EXAMPLES.  , 

x.  What  coft  823  yards,  at  D.I  29C.  per  yard  ? 

D.         D. 
823X1 -29=1061 -67  Ans. 

2,  What  coft  56  yards  2qrs.  at  D.3*i  i  per. yard;: 
56yds  2qrs.=56'5yds.  ;  and  56-5X3  11=0.175-715  .Ans, 

Before  the  queftions,  hereafter  given,  can  be  wrought, 
the  following  Tables  mult  be  perfectly  gotten  by  heart. 

TABLES. 


Pts.  of  as.  of  a 
d.         s. 

6     =     { 
4    =    i 

3     =    { 
2     =    \ 

'i-  \ 


Aliquot)  or  even  Parts  of  Money. 


-  TiS   _ 


Parts  of  a  £. 
s.    d.  £ 

10       O       =5 


£• 


TT 

i 

T7T 
i 

i 
l 
l 


M   3 


Parts  of  a 

c. 
50       = 


25 

20 


Dollar 
D. 

f 


i 


uj.  = 


T!* 

TB" 


i3» 


PRACTICE. 

or   even  Parts  of  Weight* 


Pts.ofaCwt. 

PtS.ofAC. 

Pts.ofiC.        Pts.  of  a  Ton. 

qrs.lb.  Cwt. 

Ib.     -iCwt 

Ib.  iCwt.      Cwt.qr.      T. 

2     °  =     i 

28  = 

i 

^4  =     i          10  o  — 

"g 

I       0   =      ^ 

14  = 

"<? 

7  =    i            50  = 

4: 

o   16  =     4- 

o    == 

i 

T 

I 

o  14  =    4- 

7  — 

1 

2    =    -jJ4-                  22    — 

-J 

o     8  =  TV 

4  = 

r? 

2O    = 

1 

o     7  =  rV 

1     I    = 

ITT 

°     4  ^  57 

I    0    = 

I 

Another  Table 

<?/"  aliquot  Parts  of  Mo?iey. 

Parts  of  a  fhi 

11. 

s. 

d.          ^ 

c.             D. 

d.               s 

. 

»3 

4     =       1 

80      ^      | 

30         =         4 

12 

6     =       £ 

75-4 

9=1 

12 

0       =      Tff 

7-3          =      TT 

8         = 

3 

80      =      A 

68|     =    ^ 

74-      = 

r 

7 

6=4 

/r/r<2                         L> 
0%       =           1 

.1             3 
4?          —         F    • 

6    o    =    TV 

6°t         =          ? 

Parts  of  a  Pound. 

Parts  of  a  Doll. 

56i    =     A 

s.    d.          j 

£ 

c 

D. 

43^-    =    rV 

18    o    =    - 

9 

rv 

9J*    =    W 

41T      =      A 

• 

17    6     = 

•ff 

ni  ^                      11 
9  '  ¥       —      T  2- 

40          =•         r 

16    8     = 

|- 

9 

o       —     T|f 

/,  -  i        __ 
3/2                         8 

16    o     =     - 

*TT 

8 

7?     —       "s~ 

15    o    = 

a 

?4 

8 

3"j     ~       4" 

3°       =    y35 

14   o    =    . 

rV 

8 

i  JL      —     JL  \ 

4                   16 

i  S  A     —      3 

4                   15 

A  TABLE  OF  DISCOUNT  PER  CENT; 

£ 

s.   d.  " 

£                         s.  d.  " 

I  i  per  cent. 

5=     O          3" 

I7i  Per  cent-  =36 

2-*-    — 

=  o     6; 

0 

20     —  =40 

0 

3!  _-  . 

=  09 

P 

22i  =  4     6 

s- 

5       

=  1-0 

O 

25     •               —  5     o 

a 

6J   _, 

-   i     3 

•6 

30     =60 

'  *o 

7*  

=.  i     6 

§ 

35     —      -70 

g 

^2T    — 

=  i     9 

g. 

40    =80 

P" 

10  • 

=    20 

. 

45     =90 

12^  

=    26 

50    —  —      -,JQ     o^ 

*5  

=  3    oj 

PRACTICE,  139 

CASE  I. 

When  tbj  pries  cf  lyd.  Ib.  &e.  is   an  even  part  of  one 

fin'ding  :  Find  the  value  of  the  given  quantity   at  is.   per 

yard,  Ib.  &c.  ;  then  draw  a  line  underneath  and  divide  by 

that  even  party  and  the  quotient  will  be  the  anfwer  in  fhil- 

lings,  which  mult  always  be  brought  into  pounds. 

EXAMPLES.  >., 

i.  What  will  354|yds.  coil  at,  ^d.  per  yard  ? 

s.     d 
I  td-  I  sV  I  354  6  value  of  354^yds.  at  is.  per  yard. 

Ans.  £o  7  4^  value  of  354|yds.  at  £d.  per  yard. 

Or  thus.  Or  divide  by  8  and  6,  thus, 

£    s.  d.      s.    d.  8)354  6 

8)17  14  6=354  6 

6)44  3^ 
6)2     4  3i 

7  4i  Ans. 


7  4^  Ans.  as  before. 

2.  What  will  759!  yards  come  to,  at  3d.  per  yard  ? 
3d.  |  i  |  759     9  value  at  is.  per  yard. 

2lo)i8|9   ill 
Ans.  £9  9   1  1^  value  at  3!  per  yard. 

£  s.    d. 

Or  thus.       |  3d.  |  -4-  |  37   19     9  value  at  is.  per  yd. 

Ans.  £9     9   uj-    value   of  759  yards, 
at  3d.  per  yard.  —  —  -  - 

Queftions.  Anfwers. 

yds.  £      s.     d. 

3.  642    at  Jd.  per  yard.  o     13     4^ 

4.  567i—  »id-  -  3      i0  Ili 
5-  475-i  —  4d.    -                7^5 

CASE    II. 

When  tie  price  is  pence,  and  no  even  part  of  a  /killing  s 
Find  the  value  of  the  g^ven  quantity  at  is.   per  yard 


PRACTICE. 

divide  the  pence  into  aliquot  parts,  for  divifors,  and  the 
fum  of  the  quotients  arifmg  from  them  will  be  the  an- 
fwer. 

EXAMPLES. 

i.  What  will  48 7 ,j  yards  come  to,  at  5d.  per  yard  ? 
1  3d.  ,  ?f  I  24  7     6  value  of  487^^5.  at  is.  per  yard. 

|2d.  I  £  1     6  i    io£  value  of  ditto  at  3d.  per  yard. 
4   i     3    value  of  ditto  at  2d.  per  yard. 

Ans.  £10  3     i  j  value  of  ditto  at  5d.  per  yard. 

3.  912^  yards,  at  Qd.per  yard.         Ans.  34!.  45.  4-£d« 

4.  745!  yards,  at  ud.  per  yard.       Ans.  34!.  35.  7^d. 

CASE     III. 

When  the  price  is  between  one  and  two  foiUings  :  Find 
the  value  of  the  quantity  at  i  s.  per  yard,  &c.  which  val- 
ue being  divided  by  thofe  even  parts,  which  the  pence  are 
of  is.  and  the  quotient  or  quotients  ariiing  therefrom, 
added  thereto,  the  fum  will  be  the  anfwer. 

EXAMPLES. 

i.  What  will  758^  yards  come  to,  at  is.  pd.  per  yard  I 
6d.     i     37      1 8     6     value  at  is.  per  yard. 
1 8     19     3     value  at  6d.  per  yard. 
9       9     ?i  value  at  3d.  per  yard. 


Ans.  £66       7  4-^  value  of  758!  yards,  at   is.  9d. 
per  yard. 

Queftions.  Anfwers. 

2.  793  yards,  at        iz|d.  £41     2     6| 

3.  847^     is.     id.  45   18     i£ 

4.  647!    is.     5d.  45   17     7| 

S-  752i    —  IS»  iod.  68  19     7 

CASE   IV. 

When  the  .pries  it  an  even  part  of  a  pound :  Find  the 
value  of  the  given  quantity,  at  one  pound  per  yard,  £c. 
then  draw  a  line  underneath,  and  divide  by  that  part  / 
the  quotient  will  te  the 


PRACTICE.,  141 

EXAMPLES. 

i.  What  will  156!  yards  come  to,  at  33.  4<J.  per  yard 
s.d.  £      s.    d. 

3  4  I  e  I  *56  !5  °  Price  a*  tl.  per  yard. 


Ans.  £2.6     2  6  price  at  35.  4d.  per  yard. 

Queftions.  Anfwers. 

Yds.         s.  d.  £    s.    d. 

2.  516^   at    i   o  per  yard.  25    16  9 

3.  624    —    13     39     o  o 

4.  719-4  —    i  4     .  47   19  4 

5.  648     —    i  8     '  54     o  o 

CASE     V. 

Wbtntbi  price  wants  a>!  even  part  of  a  po'und :  Firft  find 
the  value  of  the  given  quantity  at  il.  per  yard,  &c.  then 
divide  it  by  that  ci\'n  />#r^Which  is  wanting,  and  fubtracl 
this  quotient  therefrom  ;  the  remainder  will  be  the  an- 
fwer, 

EXAMPLES. 

i.  Vvrhat  will  i67-Jyds.  cod,  at  175.  6d.  per  yard  ? 
s.  d.  £      s.  d. 

|  2  6  I  g-  I   167    10  o  value  at  il.  per  yard. 

20   18  9  ditto  at  2s.  6d.  per  yard. 


Ans.  £146   1 1'  3  value  at  175.  6a.  per  yard. 

Queflions.  Anfwers. 

Yds.  s.    d.  £      s.  d. 

2-  347i  at    13  4  per  yd.  231    13  4 

3-  485J  —  15  o 364     6  3 

4.  614    —  16  o 491     40 

5.  912^  —  17  6 798     4  4j- 

CASK   VI. 

When  the  price  is  JJrilHngs,  pence,  and  farthings,  and  not 
an  even  part  of  a  pound  .-  '  Multiply  :he  given  quantity  by 
the  {hillings  in  the  price  of  i  yard,  &c  and  take  parts  of 
parts  from  the  quantity  for  the  pence,  &c.  ;  then  add 
them  together,  and  their  fum  will  be  the  anfwer  in  (hil- 
lings, &c,  Or  you  may  let  the  given  quantity  (land  as 


1 42  PRACTICE. 

pounds  per  yard,  &c.  then  draw  a  line  underneath,  and 
take  parts  of  parts  therefrom  ;  which  add  together,  and 
their  fum  will  be  the  anfwer. 

N.  B.  I  adviie  the  learner  to  work  the  following  ex- 
amples both  ways,  by  which  means  he  will  be  able  to  dif- 
cover  the  moft  concife  method  of  performing  fuch  queft- 
ions  in  bulinefs,  as  may  fall  under  this  cafe. 

EXAMPLES. 

i.  What  will  248 J  yards  coil,  at  73.  6d.  per  yard  ? 
i  6  |  i  |  2485.  fid,  value  of  248 \  yards,  at  is.  per  yard. 
7 

1739  6  value  of  ditto  at  75,  per  yard. 
124  3  value  of  ditto  at  6d.  per  yard. 

2!o)i86|3  9 
Ans.  £93  3  9  value  of  ditto  at  75.  6d.  per  yard. 

Or  thus  : 
£      s.    d. 

|  6  |  ^  j  12  86  value  of  248^-  yards,  at  is.  per  yd. 
Mult,  by  7 

&6  19     6  value  of  ditto  at  73.  per  yard. 
643  value  of  ditto  at  6d.  per  yard. 

Ans.  £93     3     9 

By  the  latter  part  of  this  cafe. 
248   10  o  value  of  248^- yards,  at  il.  per  yd. 

62     2  6  value  of  ditto  at  55.  per  yard. 
31     13  value  of  ditto  at  2s.  6d.  per  yard. 

Ans.  £ 93     3  9  value  of  ditto  at  75.  6d.  per  yard, 

Queftions.  Anfwers. 

Yds.  s.     d.  £      s.    d. 

2.  684:     at       46     per  yd.       15     8     3 

3.  124       —       58       35     2     8 

4.  146      —     14    9      107   13     6 

j.    218^    —     12     6 136  u     2 


5         ' 


2    6 


PRACTICE.  143 


CASE    VII. 

When  the  price  of  the  yard,  Ib.  Cffc.  //  pounds^  Jhillings, 
and  pence  :  Firft,  multiply  the  quantity  by  the  pounds, 
and  if  the  (hillings  and  pence  be  an  even  part  of  a  pound, 
divide  the  given  quantity  by  that  part-t  and  add  the  quo- 
tient to  the  produft  for  the  arifwer.  But  if  they  be  not 
an  even  part  of  a  pound,  you  muft  take  parts  of  parts, 
and  add  them  together  as  before.  Or,  reduce  the  pounds 
and  (hillings  into  (hillings,  and  multiply  the  quantity 
thereby,  after  which,  take  parts  for  the  pence,  and  add 
the  whole  together,  and  their  fum  will  be  the  anfwer  in 
fliillings,  &c. 

N.  B.  The  learner  (hould  work  the  following  queft- 
ions  both  ways. 

EXAMPLES. 

i.  What  will  156  yards  come  to,  at  3!.  6s.  8d.  per  yd.  I 
s.  d. 

\6  8  |  ^  |  156  o  o  value  at  il.  per  yard. 
3 


ABS.  £520  o  o 

Or  thus  : 

4111  !5<>  value  at  is.  per  yard. 

4  I  i  I    66  fliillings  in  the  price  of  i  yard. 

936 


10296  value  at  3!.  6s.  per  yard. 

52 
52 


£520  o  o 


44 


PRACTICE. 


QueQions. 
Yds. 

£  s,  d. 

Anfwers* 

£      s.  d. 

3451     at 

650 

per  yd. 

2159     7  6 

j-g-J       

36  8 

•i'     .  .. 

199     34 

75       — 

5  3  4 



387   10  o 

68       — 

460 

—  „ 

292     8  o 

CASE     VIII. 

When  the  price  of  one  hundred  weight,  &c.  is  of  fever  al 
denominations )  and  the  quantity  likcnuife  :  Multiply  the  price 
by  the  integers,  and  take  parts  for  the  reft  from  the  price  of 
an  integer  ;  which,  added  together,  will  be  the  anfwer. 

EXAMPLES. 

i.  What  will  9cwt.  3qrs.  i4lb.  of  fugar  come  to,  at 
4!.  173.  4d.  per  cwt.  ? 


qrs 

Jb.         £ 

s. 

d. 

2 

O 

1 

~rf 

4 

17 

4 

price 

of  icwt. 

I 

o 

I 

9 

O 

H 

\ 

-% 

••••  •« 

£ 

s. 

d. 

43 

16 

o 

price 

of  9 

0 

o 

2 

8 

8 

—     o 

2 

o 

I 

4 

4 

—      0 

I 

o 

0 

12 

2 

•     •»            •    •     •! 

—      0 

0 

-14 

Ans, 

-  AS    i 

2  price  oj 

Queftions. 

cwt 

.qr 

,lb. 

£ 

S. 

d. 

2. 

8 

1  6     at 

5 

17 

9 

3- 

7 

3 

19     — 

7 

12 

8 

4- 

12 

i 

24     — 

3 

18 

10 

5- 

16 

2 

17     — 

2 

15 

ii 

3  J4 

Anfwer  s. 

£  «• 

9  per  cwt.  49  8 

1 —       60  9 

—  49  2 

—  46  ii 


7t 

i 


CASE    IX. 

When  the  price  is  at  any  of  the  rates  in  the  fecond  Prac- 
tice Table  of  aliquot  parts.-  Multiply  the  given  quantity 
by  the  numerator,  and  divide  that  product  by  the  denom- 
inator ;  if  the  price  be  pence,  the  quotient  will  be  the  an- 
fwer in  {hillings  ;  if  {hillings,  the  anfwer  will  be  pounds. 


PRACTICE.  145 

EXAMPLES. 

i.  What  will  379  yards,         2.  What  will  149  yard<?, 
at  4^d.  per  yard,  come  to  ?     at  6s.  per  yard,  come  to  ? 
379  H9 

_!  — 

8)1137  il°)44l7 

2|o)i4U     i&  Ans.  £44  14 

Ans.  £7     2     i-i 

Oueftions.  Anfwers. 

s.  d.  £     s.    d. 

3.  127  yards,  at     o  7^  per  yard.     3   19     4^ 

4.  249  ditto            7  6  --       93     7     6 

5.  3,57  ditto          12  6  ----  22.3     2     6 

CASE   X. 

To  find  the  value  of  good  t  fold  by  particular  quantities  : 
viz..  1.  By  the  icore.  II.  Round  timber.  111.  By  5 
fcore  to  the  hundred.  IV.  By  112  to  the  hundred.  V. 
By  6  fcore  to  the  hundred.  VI.  By  the  great  grofs. 
VII.  By  the  thoufand, 

I.   To  find  the  value  of  Goods  fold  by  the  Score. 

The  price  of  i  is  given,  to  find  the  price  of  i  fcore. 

If  the  given  price  be  (hillings  and  pence,  or  only  pence, 
divide  the  given  price,  in  pence,  by  12.  The  quotient 
will  be  the  aniwer  in  pounds,  and  the  remainder  will  be 
fo  many  times  is.  8d. 

EXAMPLES. 

i.  At  9d.  each,   what  is         2    At  45.  9d.  each,  what 

that  per  fcore  ?  is  that  per  fcore  ? 
i2)9d.(  75=r/*o   15   oAns.  45.  9d. 

O.  t»)  inverting  thequeition.  12 

x  icore=2c=rs.  8d.  — 


*  °  j^4  'js.  Ans. 

N 


146 


PRACTICE. 


It  may  be  remarked,  that  when  the  price  is  (hillings 
and  pence,  the  anfwer  will  be  juft  fo  many  pounds  as 
there  are  {hillings,  and  fo  rflany  times  is.  8.i.  as  there 
are  pence.  If  farthings  are  given,  for  iqr.  reckon  jd. 
for  2qrs.  lod.  and  for  3qrs.  is.  3d. 

TABLE  of  Aliquot  Parts.     20  the  Integer. 


2  =  iV 

4=       T 

5  =     i 


6  = 


3 
1  CT 

4 
T(T 


12       =     T*g- 

9 

*4    ==    Tcf 


10   = 

3.  What   coft  7,   at  2s. 
9d.  per  fcore  ? 

s.    d. 
i 


18 


J5  =    i 

4.  What  coft  1 7,  at  195 
lod.  per  fccre  ? 

s.     d. 


TO- 


7= 


3? 


10 
5 

2 


19     10 

T^ 


1 6     =      1 6 


II.   Round  Timber. 

Forty  feet  make  a  load  or  ton  of  round  timber. 
If  the  given  price  of  a  foot  be  millings, 

RULE. 

Multiply  the  given  price  by  2>  and  the  product  will  b£ 
the  anfwer  in  pounds. 

5.  What  coft  a  ton,  at  33.  per  foot  ?        3s.X2=6l.  Ans. 

6.  What  coft  a  ton,  at  93.  per  foot  ?    9s.X2=i8l.  Ans. 

If  the  given  price  of  i  foot  be  pence  only,  or  fhiilings 
and  pence,  divide  the  given  price,  in  pence,  by  6.  The 
quotient  will  be  the  anfwer  in  pounds,  and  the  remain- 
der will  be  fo  many  times  35.  4d. 

7.  What  coft  40  feet,  at         8.  At  is.    pd.    per 
lyd.  per  foot  ?  what  coft  a  ton  ? 

6)17  6)21 


1  6  8 


£3   10  Ans. 


If  the  given  price  be  farthings  only,  or  pence  and  far- 
things, divide  the  given  price,  in  farthings,  by  6  ;   then 


PRACTICE.  147 

divide  that  quotient  by  4,  and  this    loft  quotient  will  be  the 
anfwar. 

9.    At    $qrs.    per   foot,         TO.  At   ij-Jd.   per  foot,, 
wh-.it  coft  a  ton  ?  what  coft  a  ton  ? 

6)3  '3i 
4 

4)0     TO  —      - 

4L_-.  6)53 

£o    *  6  Ans,  — — . 

4)8   16  8 

ft     4  2  Ans. 

Or,  Suppofe  every  (hilling  in  the  price  to  be  2!.  every 
penny  to  be  3^.  4^-  and  every  farthing  to  be  tod. 

1.1.  What  coft-40 feet,  at         ia.    What    cod    40,   at 
3qrs.  per  foot  ?  15 id.  per  foot  ? 

s.  d. 

*s.  6d.  Ans.  i  ex  2=^*2     o  o 

3  4X  3=    o  10  o 
o  ixio~    o     i   S 


£2    11   8 
III.*   To  find  the  value  of  Goods  fold  h   5  fcors   to  tbs. 

Hundred* 

i  ft.  If  the  given  price  be  pounds  and  fliillings,  or  fniU 
lings  only, 

RULE. 

Multiply  the  given  price,    in  (hillings,  by  5,    and  the 
will  be  the  anfwer  in  pounds. 


*  7*  Federal.  Money.  —  Remove  the  decimal  point  two  places  to  the 
right  for  the  ani'ver, 

EXAMPLES. 

1.  What  coft  100  yard*,  at  Da  500.  per  yard  ? 

D.z  50x1  oo=D.  250',  Ans. 

2.  What  coft  100  yards,  at   7jc.  per  yard  ? 

D.75X  100=0.75-,  Ans. 

3.  What  coft  too  yards,  at  50.  6£m.   p«r  yard  ? 

D.  05625  X  100=0.5-615,  Ans. 
•}.  What  coft  i  oo  yards,  at  370.  5m.  per  yard  ? 

Ans.  3;dols.  500. 
5.  What  coft  100  yards,  at  68c,  7|m  per  yard  ? 

Ans.  6  8  dole,  750. 


i-48  PRACTICE. 

13.  At  193.  per  yard,  14.  At  4!.  133.  per  cwt. 
what  coil  100  yards  ?  what  coft  100  cwt.  or  $ 

193.  tons  ? 

5  4     '3 

—  20 

£95  Ans-  — 

93 
5 

£4.65  Ans. 

2d.  If  the  given  price  of  i  be  pence  only,  or  fhlllings 
and  pence, 

RULE. 

Multiply  the  given  price,  in  pence,  by  5  ;  then  divide 
that  product  by  12.  The  quotient  will 'be  pounds  ;  and 
the  remainder  To  many  times  is.  8d. 

15.  If  i    yard   coft   $d.  what  coft  100  yards  ? 
9 
5 

•12)45 

£3   1S  Ans. 

16..  What  coft  100  bufliels,  at  3  ys.  4d.  per  bulhel  ? 
s,     d.  Or, 

35     4  35s-  4d- 

12  |4d.  lijj 

175 

i     13     4 


^"176     13     4 

Here  5  is  divided  by  £. 

3d.  If  the  given  price  of  i  be  (hillings  and  pence  : 
Multiply  the  price  by  5,  and  the  product  under  the  place 
of  ihillings  will  be  the  anfwer  in  pounds,  and  the  prod- 
uct under  the  place  of  pence,  will  bs  fo  many  times 
is.  8d. 


PRACTICE. 


149 


17.  At  2S.  5<I  per  bufh-         18.  At  255.   3d.  per  ton, 
el,  what  coft  ico  bufliels  ?       what  coil  ico  tons  ? 
s.    d.  s*      d. 


2     5 

5 

12        I 


25     3 
5 

126     3 


jT\2     i     8  Ans.  £126     5  Ans. 

4th.*  To  find  the  price  of  i,  at  fo  much  per  hundred 
of  5  fcore. 

GENERAL   RULE. 

Multiply  the  given  price  by    12;    divide   the    produ<fl- 
by  5>  and  ihe  quotient  will  be  the  anfwer  in  pence. 
But  if  the  price  be  pounds  only  : 

RULE. 

Divide  the  given  pries  by  5,  and  the   quotient  will  be 
the  anfwer  in  (hillings. 

19.   If  icoyds.    €0(165!.          21.  If  ioo  yards  coft  i  il. 
**hat  coft  i  yard  ?  73.  yd.  what  coft  i  yard  ? 

5)65  £     s.     d. 

—  ii     7     9 

135.  Ana*  12 


20.  If  looyds.    coft    2!.       5)^36 
1 8s.    4d.   what   is  that  per 
vard  ?  12)27 

s.    d.  — 

18     4, 

12 


£ 

2 


2s  3^d.  Ans. 
In  dividing  27  by  12  (in 
the  2ift  quellion)  the  quo- 
tient  is  2S.  and  the  remain- 
der 3  d  the  6  is  2f'o  of  a  pen- 
ny,  =  one  farthing,  and  the 
7  is  of  no  account. 

*   In  Federal  Money.  —  Remove  the  decimal  point  two  places  to  the 
•  it  for  the  anfwcr. 

EXAMPLES. 

i.  If  .100  yards  coft  D  Sjo,  what  cod  i  y^rd  ? 

D-.250 
::,  If  ioo  yards  coft  D./5,  what  coft  i  yard  ? 


N  2 


ioo=D  z-jo  Ans. 
,  ADS, 


PRACTICE. 


TABLE  of  Aliquot  Parts,      rcothe  Integer, 


5  =  V*      25 

i 

~        4 

10    =    To         30 

3 

—   Td~ 

20  =     ^-j  40 

=  rV 

22.*  At  3!.  75.  6d 

per  i< 

£ 

s.    d, 

20 

T 

3. 

-  1  1  !• 

7     6 

n 

[  2         0* 

2 

Tcf 

0 

1     4 

1      i 

0 

o     8 

23.   At  2!.    is.    red. 
*oo,  what  coft  1 8  ? 

£  ••  d- 

2     I     1O 


50    = 

60     =     /o      !     80     =     j 

70  =  Tv  I  90  =  T 

per  100,  what  will  23  coft  ? 


Add, 


Anfwer. 

24.    At    5!.    93.    6d. 
100,  what  colt  35  ? 


per 


per 


20 


TV 


59 


0    10 


7 

j 


Sub, 


10)16     8     6 


=  £0  7     6J  Ans. 


12     IO 

5     5i 


Add 


£i    1  8-     3?  Ans. 

j.  If  100  yards  coft  D  5  620.  5m.  what  coft  i  y^rd  ? 

D.5  625—  ioc=D.  O5625=rjc    6^m.  Aus,. 

4.  If   roo  y^rds  coft  D.^7  500.  what  coft  I  yard  ?     Ans     ^;t    jjm. 

5.  If  100  yards  coft  D  68  750    what  coft  i  yard  ?  Ans.  68c.  7^tn. 
*    TJ  find  the  value  cf  an\  nugtber  at  a  given    price  per  IOO  in  ftHer^ 

Mon^  —  Multiply  the  price  ptr  ZOO  by  the  given  quantity,  and 
point  off  two  right  hand  figures,  in  the  product  more  than  required 
by  multiplication  of  decimals.  Or.  point  off  the  two  right  hand 
pbces  in  the  given  quantity,  and  multiply  and  point,  as  in  multi- 
plication of  decimals 

EXAMPLES. 
I.  What  coft  56  yards,  at  D  87  yoc.  per  100  yards  .x 

D.87'5  X  5° 

-  .  -  -=D49Ans.  Or,  D  87-5  x  'S  6=0.49   as  before. 


a,  What  coft  45^ib   of  beef,  at  D.$$  per  100? 

P-5  5X45'5 

--  =D  a"5®2/=D  a  500. 


Ans- 


Or,  D.5\5X'45jU>,=D  2-5025,  as  before. 


PRACTICE.  i51 

TV.   To  find  the  value  of  Goods  fold  by  1 1 2lb.   the  C<wt. 
The  price  of  lib.  is  given  to  find  the  value  of  icwt. 

Fora  farthing,  account  2s.  4-}.  per  cwt.  F-  r  half  a 
penny,  45.  8d.  For  sqn  7-.  Ard  for  every  pen<;y  95. 
4cJ.  per  cwt. 

2?  What  coft  icwt.  at  26  At  8 jd.  per  Ib.  what 
3od-  per  cwt.  ?  cod  icwt. 

At  id.  per  Ib.    s   d.  At  id.  per  Ib.  s.  d. 

icwt.  cofts          9  4  icwt.  colls          9  4 

S  8 

At  3d.        fi     8  o|  A  jj     At  8d.       ,£3    14  8  7   . 
At  |d.  0     4  8J;  Add     At|  o     7  o]Add 

^"i    12   8   Ans.  £4     i    8  Ans. 

V.    To  find  the  value  of  Goods  fold  by  6  Score  to  the  Hundred, 
The  pi  ice  of  i  is  given,  to  find  the  price  of  i  hundred, 

R.ULE 

Suppofe  every  penny  in  the  price  to  he  fc  many  pounds, 
and  for  every  farthing,  fucn  a  part  of  .1  pound t  as  they 
are  of  a  penny  ;  then,  half  of  that  mm  will  be  the 
aniwer. 

27.  At  4^d.  per  yard,  28.  At  i6s.  g^d.  per 
what  toft  120  yards  ?  yard,  whut  coit  i  io  yards  ? 

£     s.  s.   d. 

2)4     co  16  9^- 

£*.>;••.  i 


3.  What  coft  375  yards,  at  375  dollars  per  100  yards  ? 

Ans.  D  1406  ajc, 

4.  What  coft  54  yard?,  at  16  dollars  per   100  yards? 

Ans.  D.8  640, 

y.  What  coft  51  a  yards,  at  6  dollars  25  cents  per  100  ya:iU  .' 


PRACTICE. 


To  find  the  price  of  i,  at  fo  much  per  hundred  of  6 
fcore. 

RULE. 

Multiply  the  price  by  2,  then  call  the  pounds  fo  many 
pence,  and  the  {hillings  iuch  a  pait  of  a  penny,  as  they  are 
of  a  pound,  and  you  will  have  the  anfwer. 

39.  If  120  yards  coft  3!.         30.  If  120  yards  coft  5!. 


1 8s.  6d.  what  coft  i    yard  ? 

£  s.    d. 

5   18     6 

2 


is.  whatcoft  i  yard.? 

£ 

3 


7     4 
Ans. 


TABLE  of  Aliquot  Parts.      120  the  Integer. 
Alfo, 


17     o 
.  4-  4-or.  Anre 


—  TO 


20= 


40  = 
60  = 


105  = 


45=  T 
48=  1 
50  =  fV 
70  =  rV 

31.   At  3!.  175.  6d.  per  hundred,  what  coft  14  ? 
/*     s.     d. 


72  = 
75  =    T 
80=    | 

84  =  rV 
90=    | 


96= 

100  = 


12 


3   17     <5 


o     7 
o     i 


i 
Si 


14     r=     £0     9     o^-  Ans. 

32.  At  2\.  135.  6Jd.  per         33.  At  il.    193.    3d.  per- 
hundred,  what  coft  49  ?  hundred,  what  coft  75  ? 


2   13     61 


o  17   10  o- 


d. 

3 

5 


8)9   16     3 


49  =1     i    IQ  i  Ans. 


PRACTICE.  153 

VI.    To  Jlnd  the  value  of  Goods  fold  by  the  Great  Grofs. 

NOTE.  12  make  i  dozen,  12  dozen  i  fmall  grofs,  12 
fmall  grofs  i  great  grofs. 

The  price  of  i  dozen  being  given,  in  pence  to  find  the 
price  of  a  great  grofs. 

RULE. 

Multiply  the  price  of  i  dozen,  in  pence,  by  3.  then  di- 
vide that  product  by  5,  and  the  quotient  will  be  the  an- 
fwer  in  pounds,  &c. 

For  proof  do  the  contrary. 

N.  B.  If  the  price  of  i  be  given,  the  price  of  one 
fmall  grofs  is  found  after  the  fame  manner. 

34.  What  coft  i  great    grofs,  at   i8d.  per  dozen  ? 


5)54 

£10  16 

35.  At  43.  3d.  per  doz.  what  coft  i  great  grofs  ? 
45.  3d. 

12 

JTd. 
3 


Or, 

s.  d. 
4  3 

12 

2     II     O 
12 


£30  12  Ans.  £30-12  o 


TABLE  of  Aliquot  Parts.     144  the  Integer. 
Aifo, 


12  =  AJ36  =  i 


16 

18  = 
24  = 


48  = 
72 


I  3*  =    I 

I  60  =  fV 

64=    J 

80=    i 


84  =T'f 
108  =  4 

120  =    |- 


128    =      | 

132  =  ri 


36.  At  2\.  I2S.  9d.  per  great  grofs,  what  coft  45  dozen? 

£    s.   d. 


36 

.*. 

4 

2 

12 

9 

9 

i 
4 

O 

'3 

3 

si. 

JAd 

Ans 

45 

£o 

16 

5i 

154 


PRACTICE. 


37.  What  coft  117  do*.  38,  At  3!.  i6s.  8d.  per 

en,  at  9!.  i$s.  7d.  per  great  great  grofs,  what  coft  7 

grofs  ?  great  grcfs  and  96  dozen  ? 

£  s.  d.  X    s.  d. 

9   13  7  3   16  8 


ic8 
9 


29     o  .9 


7     5 

12 


*il 


Add 


3)7  13  4 


9(5  =    2     If     l 

Top  line  X;=:26  16  8 


117     s=  £7   17  3;}-  Ans, 

VII.*  To  find  the  value  of  Goods  fold  by  the  Tkoufand. 
The  price  of  I  is  given  to  find  the  price  of  I  coo. 

RULE, 

Multiply  the  given  price,  in  pence,  by  50,  then  divide 
the  product  by  12,  and  the  quotient  will  be  the  anfwer  in 
'pounds,  &c. 

39.  At  6d.   each,   what        Or,    as   icoos.  are  50!, 


colt  1000  ?• 

6 
50 

12)300 
- 

£2$  Ans. 
40.  What  coft  1000,  at 


take  parts     for  the   pence 
out  of  50. 


Ans,  25 


each  ? 


2d. 


8  6     81 

I     2     IOJ 

£9  7     6  Ans. 


*  Tn  Federal  Money—  Remove  the  decimal  point  three  places  *o 
Che  right  or  left,  as  the  cafe  requires,  for  the  anfwer. 

EXAMPLES. 
r.     What  coft  5000  yards,  at  5  cents  per  yard  ? 

•05x1000=5 050-^0.50,  Ans*. 


PRACTICE. 


VIII.*    To  jind  tbs  price  of  one  at  fo  much  per  tboufand. 

RULE. 

Multiply  the  price  by  12;  divide  the  producl  by  50  j 
then  take  the  pounds  for  fo  many  pence,  and  the  ihil- 
lings  for  fuch  a  part  of  a  penny  as  they  are  of  a  pound, 
which  will  be  the  anfwer.  £,  s.  d. 

At  5!.  45.  2d.  per  1000,  542 

•A'hat  coft  i  ?  12 


50 


{ 


5)62   10     o 

!0)I2     10 


42.  At  354!.  35.  4d.  per 
1000,  what  coft  i  ? 

£       «•    d- 

354     3     4 

12 


10)4250     o     o 

5)425 


Ans. 


Or, 


s.    d. 


100 
10 

I 

rV 

354     3     4 

35     8     4 

3    10  10 

Ans.  071 


Ans.  75.   id. 
TABLE  of  Aliquot  Parts.     1000  the  Integer. 


5°  = 


Alfo, 

300  =  fa 

700  =  Tv 

375  =    4 

75©  =    -1 

400  =    § 

800  =    •£ 

600  =   4 

875  =    i 

625  =    4 

^oo  =  fa 

T(T 
125    =       1 

2OO  =  3- 
250  =  i 
500  =  -1- 

a.  What  coft  IOQO  yards,  at  iz  cents  5  mills  per  yard  ? 

Ans.  D.I 25, 

3.  If  1000  yards  coft  0.37  500.  what  coft  i  yard  ? 

D.37*5-f-i ooo=D. "0375=30.  'i ^;fn.  or  34^*  Ans. 

4.  If  icoo  yards  coft  D.i6z5,  what  coft  i  yard  ? 

Ans,  D.I  6ac.  jm, 
*  See  the  preceding  Note. 


PRACTICE. 

t  il.  f7s,  9d-  per  i  ceo,  what  coft  115  ? 

£  s    d. 

100  I  TIT     i    *7     9 

*°  ,  T'U     o     3     9fJ 

.o     o     4^  V  Add. 
5  I    T  i  o    o    2iJ 

115     =  £o     4     4  Ans. 

44.  At    2\.    is.   8d.    per   xoco,  what  coft  875  i? 
£  s-  d. 

2        L    8 

7 
8)14   n   8 

£1    1 6   <;t   Ans. 
"45.  What  coft  33,  at  245-  8d.  per  loco'? 


5° 

p 

i     4 

^5 

2 

O       1 

•AMMMM— 

0       0 

5 

I 
T 

O       O 

30 

n: 

O       O 

3 

rV 

0      0 

33 

= 

o     o 

7f  I  Add. 
^3 


Add. 


9i 


*  7*o  fi>'d  the  value  of  say  number^  at  a  given  price  per  tboufand*  itt 
federa>  Mvnev. — Multiply  the  price  per  1000  by  the  given  quantity, 
and  point  off  three  right  ha,.d  figures  in  the  product  more  thc.ii  re- 
quired by  multiplication  of  decimals.  Or,  point  ofFthe  three  right 
hand  places  in  the  ^iven  quantity;  and  multiply  and  point  as  in 
muJtiplicatiomot  decimals 

EXAMPLES. 

1.  What  coft  8;  5,  at  D  f3  per  1000. 

875 x  rj$—   i ; '-:  :  and  1 137 5-7-1  ooo=r  ''375=0.11  37c.5m.  Ans. 

2.  Wlv.it  coft  39175  feet  of  hoards,  at  D.i6  ptr  loco? 

A!. s   D.6a6  8oc. 

3.  What'coft  3x5  nails,  at  D.I    jo  cents  per  ioco?       .    « 

Ans.  4oc.  ^*m."  or,  48*0* 


v?  \CTICE. 


CASES  IN  FEDERAI  r-T. 

CASE    I. 

/F/:f«  the  price  of  l  it  an  aliquot  part  of  a  dollar.  —  Divide 
the  quantity  by  the  denominator  of  die  fraction,  wh.ch 
the  price  is  of  a  dollar,  and  the  quotient  will  be  the  an- 
fwer  in  dollars,  £c. 

EXAMPLES. 

j.  What  coft  227  yards,  at  50  cents  per  yard  ? 
c.    D      D. 
50  |  -£-  |  22y=price  at  D.I  per  yard. 

D  1  13   500.  Ans. 

2.  What  coft  265  yards,  at  i2c.  5m,  per  yard  ? 

Ans.  D  33   i2c.  5m. 

3.  What  coft  269!  yards,  at  i6|c.  per  yard  ? 

£  I  269-5  price  at  D.I. 


D-44  9  ic.  7m.  Ans. 
4.  What  colt  1050  yards,  at  6~c.  per  yard  ? 

Ans.  D  65  62c.  5m. 

CASE   II. 

When  the  price  of  I  is  two  or  more  aliquot  parts  of  a  dol- 
lar added  together  :  Divide  the  given  number  tirit  for  one 
aliquot  .part,  then  for  another,  &c.  the  quotients  added 
together,  will  be  the  anfwer. 

EXAMPLES. 

I.  What  coft  298  yards,  at  75  cents  per  yard  ? 
c.    D      D. 


5° 
25 

i 

2 

1 

4 

298 

price 

ditto 
ditto 

atD 

at 
at 

i 

149 

74 

50 

•50 
-25 

Ans.   223  50  ditto  at         '75 

2.  What  coft  927  yards,  at  53  jC.  per  yard  ? 

Ans.  D-494  400. 

3.  What  coft  6t8  yards,  at  87^0.  per  y.ird  ? 

Ans.  D.540   75c. 

4.  What  coft  328  yards,  at  570,  5m.  per  yard  ? 

Ans,  D,i88  6oc. 
O 


PRACTICE. 


CASE   III. 

When  the  price  of  t  is  the  difference  between  two 
farts  of  a  dollar  ;  Find  the  price  at  the  greater  aliquot 
part,  and  then  at  the  lefs,  and  their  difference  will  be 
die  aniwer. 

EXAMPLES. 

j.  What  coft  328  yards,  at  13!  cents  per  yard  ? 
c.      D.  D. 


331 


20 


price  at  D. 


109  33-3-     ditto  at 
65  60       ditto  at 


•20 


Ans.  43  73^     ditto  at 

2.  What  coft  817  yards,  at  3oc.  per  yard  ? 

Ans    D  245  loc. 

3.  What  coft  296  yards,  at  15  cents  per  yard  ? 

Ans    D.44  400. 

CASE  IV. 

When  the  price  cf  l  is  any  fum  lefs  than  a  dollar  :  Divide 
the  given  price  into  aliquol  parts,  either  of  a  dol;  r,  or 
of  each  other  :  find  the  price  at  each,  and  add  them  to- 
gether for  the  anfwer. 

EXAMPLES. 

i.  What  coft  279  y  mis,  at  31  cents  per  yard  ? 

price  at  U.  i 


ditto  at 
ditto  at 
ditto  at 


25C. 

5 
i 

3  ic, 


t.  What  coft  953  yards,  at  570  per  yard  ? 

Ans.  D-543  2ic. 
3.  What  coft  839  yards,  at  36c.   per  yard  ? 

Ans.  0.302  40. 

CASE    V. 

When  the  price  of  i  is  any  fum  between    i  and  2  dollars  : 
The  quantity  itielf  in  dollars,  is  the  price  at  D.  i  ;  then 


PRACTICE.  159 

finding,  by  the  preceding  rules,   the  price  at  the  parts_of 
D.I,  the  fum  of  the  whole  is  the  anfwer. 

EXAMPLES. 

i.  What  coft  386  yarcis,  at  D  1-650.  per  yard  ? 
386  price  at   D.  i 


iD. 


IO         j    -j  of  JOC. 

3"      1  £of  10 


193  ditto  at  500. 

38  60       ditto  at  10 

19   30       ditto  at  5 


Ans.  0.636  90       ditto  at   D,  i    650. 

2.  What  coil  849  yards,  at  D.  1*72  pei  yard  ? 

Ans.  D  1460-28. 

3.  What  coft  294  yards,  at  D.  i'i8  per  yard  ? 

Ans.  D.34'2'92. 

CASE    VI. 

When  the  price  of  i  is  any  number  of  dollars  and  parts  of 
3  dollar :  Multiply  the  quantity  by  the  number  of  dol- 
lars ;  and,  finding,  by  the  preceding  rules,  the  price  at 
the  parts  of  D.  i,  the  mm  of  the  \\hok  is  the  anfwer. 

EXAMPLES. 

T.  What  coft  395  yards,  at  D  3  24C.  per  yard  ? 
c.  D.      c. 

20  I  -]D.  395  price  at  D.I 

!        3  

4  I  ^  of  2oc.  I  1 185  ditto  at       3 

79  ditto  at  2oc. 

15   80       ditto  at  4 

Ans.  0.1279  80       ditto  at  D. 3  24c. 

z.  What  coft  269  yards  at  D.2  6oc.  per  yard  ? 

Ans.  D.699  400. 

3.  What  coft  694  yards,  at  D.I 2 -lo  per  yard  ? 

Ans,  D.8?97  400. 

4.  What  coft  318  yards,  at  D  4*125  per  yard  ? 

Ans    D.I 3 u   75C. 

5.  What  coft  175  yards,  at  04-44  Per  7^  -? 

Ans.  D-777* 


160  PRACTICE. 

CASE    VII.* 

When  the  price  of  i  contains  the  fame  aliquot  part  of  a 
dollar  any  number  cf  times  exafth  ;  or,  in  o+h?r  iuords,  ivhen 
the  price  of  \  has  an  aliquot  part,  which  is  aJfo  an  aliquot 
part  of  a  dollar  :  Fit  ft,  find  the  value  of  the  given  quan- 
tity at  the  aliquot  part  ;  then  multiply  this  by  the  num- 
ber of  times,  which  the  aliquot  part  is  contained  in  the 
given  fum,  for  the  anfwer. 

Or,  Since  the  price  in  this  cafe  is  always  fuch  a  num- 
ber, as,  being  divided  by  the  aliquot  part,  wili  make  the 
numerator  of  a  fraction,  of  which  the  denominator  is  the 
denominator  of  ihat  fraction,  which  the  aliquot  part  is  of 
a  dollar  :  Multiply  the  quantity  by  the  numerator,  and 
divide  the  product  by  the  denominator,  (or,  when  con- 
venient, divide  the  quantity  by  the  denominator,  and 
multiply  the  quotient  by  the  numerator,)  for  the  anfwer. 

EXAMPLES. 
i.  What  cofl  384  yards,  at  87^  cents  per  yard  ? 

.=^  of  -875.=  j  -J-D.  |  384*=price  at  D.I 


at         *i2 
X  7  X  7 


Ans.  D.336'=ditto  at 

Or  thus, 

384X7 
•875=D.-J,  and  384XJ==i-|---(==3|4x7)=D.336  anfwer  a$ 

before. 

2.  What  coft  842  yards,  at  66|c.  per  yard  ? 

Ans.  D  56  (   33|c. 

3.  What  coft  912  yards,  at  55  cents  per  yard  ? 

Ans.  D.joi  6oc. 

•MISCELLANEOUS  QUESTIONS. 

1.  What  coft  300  yards,  at  27  cents  per  yard  ? 

Ans.  D.8r. 

2.  What  coft  917  yards,  at  D.  1-125  Per  7ard  • 

Ans.  D.I03I  62c.  5m. 

3.  What  coft  35y  yards,  at  35  cents  per  yard  ? 

Ans.  D.izSzD- 


TARE  AND  TRET.  161 

4.  What  cod  8625-  feet  of  boards,  at  D.iz  per  M  ? 

Ans.  D.  10  340.  6m. 

5.  What  coft  32159ft.  boards,  at  0.13-75  per  M. 

Ans.  D-442  i8c.  6^m. 

BILL  OF   PARCELS. 

Newburyport,  Jan.  ift,   1809, 
Mr.   Timothy  Hudjler 

Bought  of  Samuel  Merchant, 
.  Bohea  Tea,  at  gs.  6d.  per  Ib. 
48  ih.  Cheefe,  at  pd.  per  Ib. 
15   Pair  Worfted  Hofe,  at  53   8d.  per  pair, 
+  :,  Dozen  Women's  G'oves,  at  36^*  6d.  per  dozen, 
19  Dozen  Knives  and  Forks,  at  55.  yd.   per  dozen, 
9  Grindftones,  at   153.  (,-].  per  Stone, 
4   C\vt.  Brown  Sugar,  at  515   per  cwt. 
31  Ib.  Loaf  Sugar,  at  is.  ojd.  per  Ib. 

Received  payment  in  full, - 

Samuel   Merchant. 


TARE  AND    TRET. 

TARE  and  Tret  are  practical  Rules  tor  deducing  cer» 
allowances,  which  are  made  by  merchants  and  tradef- 
men  in  felling  their  goods  by  weight. 

Tare  is  an  allowance,  made  to  the  buver,  for  the  weight 
of  the  box,  barrel,  or  bag,  &c.  which  contains  the  .goods 
bought,  and  is  either  at  fo  much  per  box,  &c.  at  fo  much 
per  cwt.  or  at  fo  much  in  the  grofs  weight. 

Tret  is  an  allowance  of  4lb.  in  every  io4lb.  for  wade, 
duft,  &c. 

Cloff  is  an  allowance  of  zlb.  upon  every  3cwt. 
Grofs  weight  is  the  weight  of  any  fort  of  goods,  togeth- 
er with  the  box,  buTsl,  or  big,  Sec.  which  contains  them. 
Suttle  is  when  part  of  the  allowance   is  deduced  from 
the  grofs. 

Ne<tt  weight  is  what  remains  after  all  ^allowances  are 
made. 

O  a 


TARE  AND  TRET. 

CASE    I-/* 

When  the  tare  is  at  fo  much  per  box>  barrel,  or  bag) 
Muit.pi;   ihe  nuzubei    otf    ,.>;;xes,  bantc;      ex 
and  iuotrad  the  product  irom  the  grots,  and  the 
der  wiii  be  the  neat  weight  required. 

EXAMPLES. 

I.  In  6  hogfheads  of  iugar.  each  weighing  90 wt.  2qrs. 
lolb.  grois,  tare  2 jib.  per  hogfliead  ;  how  much  neat  I 

Cwt.qr.  ib.      Cwt  qr.    Ib. 

25X6-1    i    10         92      i  o  grofs  weight  of  i  hhd. 

6 

57     2       4  grofs. 
i      i      10  tare. 


Ans.  56     q     22  neat. 

2.  In  5  bags  of  cotton,  marked  with  the  grofs  weight 
as  follows,  tare  23)6.  per  bag,  what  neat  weight  ? 

Cwt.  qr  Ib. 
A'=  7  i  19 
B  «  3  3  27 
C  =  5  i  12 
D  =  6  o  15 
E  =  8  i  c 

Ans.  3ccwt.  oqr,  T4lb.  neat. 

3.  What  is  the  neat  weight  of  15  hogiheads  of  tobac- 
co, each  weighing  7cwt,  iqr.  i3ib.  ure  looib  per  hhci.  ? 

Ans.  97cwt.  oqr.  nib. 

CASE    II. 

When  the  tare  is  at  fo  much  per  civt.  :  Divide  the  grofg 
weight  by  the  aliquot  parts  ot  a  cvvt.  iubtracl  the  quotit-nt 
from  the  grofs,  and  the  remainder  will  be  the  neat  weight. 

EXAMPLES. 

I.  In  I29cwt.  sqrs,  i6ib.  grofs,  tare  5410.  per  cwt. 
what  neat  weight  ? 

*  This,  as  well  as  every  ether  cafe  in  this  ruic,  ia  only  an  appli- 
cation of  the  rules  ot  Proportion  and  Practice. 


TARE  AND  TRET.  163 


Cvt    qi. 
i4lb.  I  |     129     3 
j  ,         16     o 

ib. 
1  6  grofs, 
26^  tare. 

Ans     113      2 

1  7  £  neat. 

2.   In  97cv/t.    iqr,  ylb. 

i    neat  v  e;,Jir  ? 
Cwt.  qr.  Ib. 
16     i      97     i     7 

grofs,  tare  2olb,  per  cwt.  what 
grofs. 

44      *3     3   i? 
3     i   25 

|  Add, 

Subtnuft   17     i    14 

tare, 
neat. 

Ans.   79     3   21 

3.  What    is  the  neat  weight  of  9   barrels  of  pot-afli3 
each  weighing  sojib.  grofs,  tare  i2.U>.  per  cwt.  ? 

Ans.   2450!!).   1402.  4-3 dr. 

4.  What  is  th?  valu^  of  the  neat  weight  of  7  hogiheads 
of  tobacco,  at   5!.   75.  6J.  per  cwt.   each  weighing  bcwt. 
3c|rs.  lolb.  grofs,  tare  2i'b    per  cwt  ? 

/ins.    270!.  45.  4^d.  reckoning  the  odd  ounces. 

CASE  III. 

When  trst  if  allowed  with  tare  :  Divide  the  futtle  weight 
by  26,  and  the  quotient  will  be  the  tret,  which  fubtraft 
fiom  the  futtle,  and  the  remainder  will  be  the  neat. 

EXAMPLES. 

i.  In  247cwt.2qrs.  i5lb.  grofs,  tare  2 Bib   per  cwt,  and 
-Ub.  ior  every   10410.  what  neat  weight  ? 

Cwt.  qr.    Ib.     oz 
I  28  ]  £  j  247      i      15         grofs. 

6 1      3      17     12  tare  fubtracl. 

1  4  I  A  i  l85     2     25       4  Aittle. 

7     o     1 6       o  tret,   fubtra&* 


Ans.    178     2       9       4  neat, 


1  64  INVOLUTION. 

2.  What  is  the  neat  weight  of  4  hog  {heads  of  tobacco? 
weighing  as  follow  : 

The  ift  5cwt.  iqr.  i2lb.  grofs,  tare  6$\b. 
2d   3         o       19  75 

3d  6         3       oo  49 

4th  4         29  35 

and  allowing  tret  to  each  as  ufual  ? 

Ans.   lycwt.  cqr. 


CASE   IV. 

When  tare,  tret,  and  doff  are  allowed  :    Deduct  the   tare 
and  tret  as  before,   and  divide  the  futtle  by  168,  and  the- 
quotient  will  be  the  cloff,  which  fubtracT:   from  the  futtle, 
and  the  remainder  will  be  the  neat. 
EXAMPLES. 

i.  What  is  the  neat  weight  of  ihhd.  of  tobacco,  weigh- 
ing i6cwt.  2qrs.  2olb.  grofs,  tare   i4lb.  per  cwt.  tret  4lb., 
per    104,  and  cloff  2lb.  per  3cwt.  ? 
J4ib,  is  -g-)i6     2     20     o  grofs. 

20       98  tare  fubtracl,, 


.  is  j?V):4     *     I0     8 

2       6  13  tret,  fubtracl. 


2-lb.  is  T67)H     o       3   ir  futtle. 

o     o       95  cloff,  fubtracT. 


Ans.  13     3     22     6  neat. 

2.  If  9  hogiheads  of  tobacco  contain  85cwt  oqr.  2lb, 
tare  ^olb  per  hogfliead,  tret  and  cioff  as  ufual,  what 
will  the  neat  weight  come  to,  at  6fd.  per  Ib-  after  deduct- 
ing for  duties  and  other  charges  51!.  i  is.  8d.  ? 

Ans.      18     i8s.    d. 


INVOLUTION, 

OR  TO  RAISE  POWERS. 

A  POWER  ii  the  produft  ariftng  from  multiplying 
any  given  number  into  itfetf  continually  a  certain  num- 
ber of  times,  thus ; 


INVOLUTION.  165 

3X3  =  9  is  the  2d  power,  or  fquare  of  3          =3* 
3X3X3  — -7  's  ^ie  3d  power  or  cube  of  3        |    =  3" 
3X3X3X3  =  81  is  the  4th  power,  or  the   biquacirate 

of  3,  &c.  =  34 

The  number  denoting  the  power  is  called  the  indsx,  or 
the  exponent  of  that  power.  Thus,  the  4th  power  of  3  is 
8 1,  or  34  ;  tli?  fecond  power  of  5  is  25,  or  5-,  &c. 

2X2=4,  the  fquarc  of  2  ;  4X4=1 6= 4th  power  of  2  ; 
i6xi6--256=8th  power  of  2,  &c. 

RULE. 

Multiply  the  given  number,  root,  or  firft  power,  con- 
tinuilly  by  itfeif,  till  the  number  of  multiplications  be  I 
lefs  than  the  index  of  the  power  to  be  found,  and  the  laft 
product  will  be  the  power  required. 

NOTE.  Whence,  bacaufe  fractions  are  multiplied  by 
taking  the  products  of  their  numerators,  and  of  iheir  de- 
n.n v.inators,  they  will  be  ipvired  by  laifing  each!  of  their 
ter:r»s  to  the  power  required,  and  if  a  mixed  number  be 
propnfed,  either  reduce  it  to  an  improper  fractions  or  re- 
duce the  vulgar  fraction  to  -J,  and  proceed  by 
the  rule. 

EXAMPLE?, 

I .  What  is  the  5th  power  of  9  ? 

9 
9 

8 1 25  id  poorer, 
9 

7 2 9= 3d  power. 


6561=4^  power. 
9 

59049= 5th  power,  or  ?4.r.fwer=9  . 
z.  What  is  th.  5th  power  of  |  ?  Ans. 

3.  What  is  the  4th  power  of  '045  ? 

Ans.  -000004100625. 
Here  we  fee,   that   in  raifmg  a  fradicn   to   a   higher 
power,  we  decreafe  its  value. 


INVOLUTION. 


CO 

*^» 

I 
§ 

PQ 

H 
< 


oo 

I 

CO 

CO 

\ 

1 

531441 

4782969 

CO 

s 

CO 

387420489 

34867844011 

313810596091 

282429536481] 

30 

IS 

CM 

cc 

00 
CO 

co 

262144 

2097152 

16777216 

CO 

r  . 

CM 

CO 

1073741824 

8589934592 

68719476736 

CO 
CO 

1 

i 

CD 

117649 

823543 

5764801 

40353607 

282475249 

197732674H 

13841237201 

CO 
CO 

CO 

.S 

CO 

CO 

CO 
CO 

98661,2 

1  2 
CD 

Q 

CO 

10077696 

60466176 

362797056 

2176782336 

CN 

o 

CM 
00 

CN 

00 

CN 

CO 

o 
A 

CO 

1953125 

9765625 

48828125 

«5 

CO 

CO 

CO 

10 

CN 

§ 

CC 

CO 

CO 
CO 

262144 

1048576 

1 

e> 

CO 

Cl 

cs 

00 

2 

CM 

F- 

03 

CD 

CO 

00 

cr> 

i 

UN 

00 

CO 

CN 
CO 

CC 

00 

CD 
CN 

CM 

CN 

O 

cc 

CO 
0 

^ 

~c?r~ 

-a-| 

i 

XO 

CO 

^ 

co    j 

O 

O 

^ 

1 

CB 

1 

1 

1 

d 

s 

OJ 

'§ 

ISurfolids 

CU 

3 

u 

g- 

rt 

•Second  Surfolids 

1 

3 

Ji 

jCubes  cubed 

ISurfolids  fquared 

JThird  Surfolids 

J£qua.  (Cube  fquared 

EVOLUTION.  167 

EVOLUTION, 

OR  THE  EXTRACTION  OF  ROOTS. 

THE  Root  is  a  number,  whofe  continual  multiplica- 
tion into  itielf  produces  the  power,  and  is  denominated 
th  iquare,  cube,  btqiudrate,  or  2d,  3d,  4th,  &c  root,  ac- 
cordingly as  it  is,  when  raifed  to  the  zd,  36,  &c.  power, 
equ.il  to  that  power.  Thus,  4*5  the  fquare  root  of  16, 
becauie  4X4=16,  and  3  is  the  cube  root  of  27,  becaufe 
3X3X3=27,  and  fo  on. 

Although  there  is  no  number  of  which  we  cannot  find 
any  power  exactly,  yet  there  are  many  numbers,  of  which 
preciie  roots  can  never  be  determined.  But  by  the  help 
of  decimals,  we  can  approximate  towards  the  root,  to  any 
affigned  degree  of  exa&nefs. 

The  roots,  which  aproximate,  are  called  fur  -d  roots  ;  and 
thole  which  are  perfectly  accurate,  are  called  rational  roots. 

Roots  are  fometimes  denoted  by  writing  the  character 
A/  before  the  power,  with  the  index  of  the  power  over  it  ; 

thus  the  3d  root  of  36  is  expreffed  \/  36,  and  the  2d  root 
of  36  is  -v/  36,  the  index  2  being  omitted  when  the  fquare 
root  is  deligned. 

If  the  power  be  expreffed  by  feveral  numbers,  with 
the  fi^n  +  OT  --  between  them,  a  line  is  drawn  from  the 
top  of  the  figri  over  all  the  parts  of  it.  Thus  the  $d 

root  of  47+22  is    ^47+22,  and  the  zd  root  of  59—17 


s 
Sometimes  roots  are  defigned  like   powersi  with   frac- 

A 

tional  indices.     Thus  the  fquare  root  of  15   is    15*,  the 

cube  root  of  21  is  21  T,  and  the  4th   root  of  37-  —  20  is 
_  x 

37—  zo4,  &c. 


i68  ...  ;iOS    OF  THE 


EXTRACTION  OF  THE  SQUARE  ROO<T. 

RULE. 

1.  Diftingulfli   the   given    number    into    periods   of 
two  figures   each,  by  putting  a  point  over  the   place  of 
units,  another  over   the  place  of  hundreds,  and   fo  on, 
which   points  (hew  the  number  of  figures  the  root  will 
confift  of. 

2.  Find  thegreateft ?  fquare  nnmber  in  the  firft,   or   left 
hand  period,  place  the  root  of  it  at  the  right  hand  of  the 
given  number,  (  after  the  manner  of  a  quotient  in  divifion) 
for  the  firft  figure  cfihe  root,  and  the  fqu^re  number  under 
the  period,  ;r-;d  fubract  it  therefrom,  s.nd  to  the  remainder 
bring  down  the  next  period  for  a  dividend. 

3  Place  the  double  of  the  root,  already  found,  on  the 
left  hand  of  the  dividend  for  a  divifor. 

4.  Seek  how  often  the  divifor  is  containedin  the  dividend, 
(except  the  right  hand   figure)  and  place   the   anfwer  in 
the  root  for  the  fecond  figure  of  it,    and   likewife    on  the 
right  hand  of  the  divifor  :  Multiply  the  divifor,   with    the 
figure  Uft  annexed,  by  the  figure  1  aft  placed  in  the  root, 
and  fuhtracT:    the  product  from    the    dividend  :    To   the 
the  remainder  join  the  next  period  for  a  new  dividend. 

5.  Double  the  figures  already  found  in  the  root,for  anew 
divifor,  ( or  bring  down  your  lafl  divifor  for  a  new  one, 
doubling  the  right  hand  figure  of  it)  and  from  thefe  find 
the  next  figure  in  the  root  as  laft  directed  ;    and   continue 
the  operation  in  the  fame  manner  till   you  have  brought 
down  all  the  periods. 

NOTE  i.  If  when  the  given  power  is  pointed  off  as 
the  power  requires,  the  left  hand  period  fhould  be  defi- 
cient, it  muft  neverthelefs  ftand  as  the  firft  period. 

NOTE  2.  If  there  be  decimals  in  the  given  number,  it 
mu 'I  be  pointed  both  ways  from  the  place  of  units  :  If 
when  there  are  integers,  the  firft  period  in  the  decimals  be 
deficient  it. may  be  completed  by  annexing  fo  many  cyphers 
as  the  power  requires  :  And  the  root  muft  be  made  to 
corfiftoffo  many  whole  numbers  and  decimals  as  there 
are  periods  belonging  to  each  ;  and  when  the  periods 
belonging  to  the  given  number  are  exhau'led,  the  opera- 
tion may  be  continued  at  pkafure  by  annexing  cyphers. 


SQUARE  ROOT. 

EXAMPLES. 
i»  Required  the  fquare  root  of  30138696025. 

30138696025(173605  the  root. 

i 

*ft  Divifor=27)20i 
189 


2d 

1029 

3d  Divifor=3466)  20969 
20796 

4th  Divifor=347205)  1  736025 
1736025 


;:.  Required  the  fquare  root  of  575*5. 

575'5°(23'98  -f 
4 


129 

469)4650 
4221 

4788)42900 


4596  Remainder. 

3.  What  is  the  fquare  root  of  10342656  ? 

Arts.  3216. 

4.  What  is  tfie  fquare  root  of  964*5  192  360241  ? 

Ans.  3.  -05  67  1. 

5.  What  is  the  fquare  rtot  of  '0000316969  ? 

Ans.  '00563, 
P 


i7o  SQUARE  ROOT. 

NOTE.  When  more  than  half  the  root  is  found,  the  re^ 
mainit:g  figures  of  it  may  be  found  by  Divifion,  making 
tiff  of  the  lail  divifor,  and  bringing  down  fo  many  of  the 
next  figures  of  the  refolvend,  as  there  are  periods  to  come 
down  when  you  began  the  divifion. 

RULES  for  the  Square  Rovt  r.f  Vulgar  Fractions  and  Mixed 
Numbers, 

A£er  reducing  the  fraction  .to  its  lowed  terms,  for  this 
and  all  other  roots  ;  then, 

i  Extract  the  root  of  the  numerator  for  a  -nevj  nume rator> 
and  the  root  of  the  denominator  for  a  nenv  denominator^ 
which  is  the  beft  method,  provided  the  denominator  be 
a  complete  power  But  if  it  be  not, 

2.  Multiply  the  numerator  and  denominator  together  ; 
and  the  root  of  this  product  being  made  the  numerator 
to  the  denominator  of  the  given  fraction,  or  made  the 
denominator  to  the  numerator  of  it,  will  form  the  fraction- 
al part  required  :  -—Or. 

3  Reduce  the  vulgar  fraction  to  a  decimal,  and  extract 
its  root. 

4.  Mixed  numbers  may  either  be  reduced  to  improper 
fractions,  and  extracted  by  the  firft  or  fecond  rule  ;  or  the 
vulgar  fraction  may  be  reduced  to  a  decimal,  then  joined 
to  the  integer,  and  the  root  of  the  whole  extra&ed. 

EXAMPLES, 

i .  What  is  the  fquare  root  of  T^ff ?  ? 
By  Rule  i. 

— r<5~irT         1<r>(4  root  °*  t^le  numerator. 
16 

1681(41  root  of  the  denominator, 

T6 

.8 1 )  8  e     Therefore,  ¥4r  =  the  root  of  the  given  frao 
81  tion. 

By  Rule  2. 
i  6X1 68 1 =26896,  and  ^26896=164.     Then 


SQUARE  ROOT.  171 

By   Rule  3. 
1 68  0 1 6( -0095 1 8  [439+     And  ^009? '8143 9=^097 5 6-K 

2.  What  is  the  fquare  root  of  |-JJ£  ?  Ans.   T"2-. 

3.  What  is  the  fquare  root  of  42^  ?  Ans   6{. 

NOTE.  In  extracting  the  fqaare  or  cube  root  of  any  iurd 
number  there  is  always  a  remainder  or  fraction  left  when 
the  root  is  found  :  To  rind  the  value  of  which,  the  corn- 
nion  method  is  to  annex  pairs  of  cyphers  to  the  refolvend 
for  the  fquare>  and  ternaries  of  cyphers  to  that  of  the  cube, 
which  makes  it  tedious  to  difcover  the  value  of  the  remain- 
der, efpecially  in  the  cube.  Now  this  trouble  may  befavsd 
by  the  following  method. 

Ini  the  fqaare  the  quotient  h  aiways  doubled  for  a  new 
divifor  :  Therefore  when  the  work  is  completed,  the  root 
doubled  is  the  true  divifor,  or  denominator*  to  its  owu 
fraction;  as,  if  the  root  be  12,  the  denominator  will 
will  be  24  ;  to  be  placed  under  the  remainder  ;  which  vul- 
gar fraction,  or  its  equivalent  decimal,  mult  be  annexed 
to  the  quotient,  or  root,  to  complete  it. 

If  to  the  remainder,  either  oi  the  fquare  or  cube,  cy- 
phers be  annexed,  and  divided  by  their  refpective  denomi- 
nators, the  quotient  will  produce,  the  decimals  belonging 
to  the  root. 


APPLICATION  AND  USE  OF  THE  SQUARE 
ROOT. 

PROBLEM  I.  To  find  a  mean  proportional  between  twz 
numbers. 

RULE.  M-altiply  the  given  numbers  together,  and 
extract  the  fquare  root  of  the  product ;  which  root  will  be 
the  mean  proportional  fought. 

EXAMPLE. 
What  is  the  mean  proportional  between  24  and  96  ? 

-^96X24=48  Anfvver. 

*  Thefe  denominators  give  a  f mall  matter  too  much  in  the  fquare 
root,  and  tooiittJc  in  the  cu'jc,  yet  they  will  be  fufficient  in  com- 
mon ufe. 


J7i  SQUARE  ROOT. 

PR  OB.  II.  To  find  the  fide  of  a  fquare  equal  tn  area  to 
any  given  fupeificits  whatever • 

RULE.  Find  the  area,  and  the  fquare  root  is  the  fide 
of  the  iquare  fought. 

EXAMPLES. 

1.  If  the  area  of  a  circle  be   184-125,  what  is  the  fide 
of  a  fquare  equal  in  area  thereto  ? 

-v/ 1 84'.  25=  i  ^-5694-.  Anfwer. 

2.  If  the  area  of  a  triangle   be   160,  what  is  the  fide 
of  a  fquare  equal  in  area  thereto  ? 

*/ 1 60=  i  2*  6494-  Anfwer. 

PROB.  III.  A  certain  general  has  an  army  of  5625 
men  :  Pray,  how  many  mud  he  place  in  rank  and  file, 
to  form  them  inf.o  a  fquare  ?  ^$625=75  Ans.* 

PROB.  IV.  Let  1052  men  be  fo  formed,  as  that  the 
number  in  rank  m^y  be  double  the  file. 

y'lojLl?  __^  -m  £iCj  ancj  74x2=148   in  rank. 

PROB.  V.  If  it  be  required  to  place  2016  men  fo  as 
that  there  may  be  56  in  rank  and  36  in  file,  and  to  ftand 
4  feet  diftartce  in  rank,  and  as  much  in  file  ;  how  much 
ground  do  they  (land  on  ? 

To  anfwer  this,  or  any  of  the  kind,  ufe  the  following 
proportion  : — As  unity  :  to  the  diftance  ::  fo  is  the  num- 
ber in  rank  lefs  by  one  :  to  a  fourth  number  ; — next  do 
the  fame  by  the  fi^e,  and  multiply  the  two  numbers  togeth- 
er, found  by  the  above  proportion,  and  the  product  will  be 
';he  anfwer.  -\ 

As  i  :  4  ::  56 — r  :  220.  And,  as  i  :  4  ::  36 — i 
t  140.  Then,  220X140=30800  fquare  feet,  the  anfwer. 

FROB  VI.  Suppofe  I  would  fet  out  an  orchard  of 
600  trees,  fo  that  the  length  fliall  be  to  the  breadth,  as 
3  to  2,  and  the  diftance  of  each  tree,  one  from  the  other, 

*  If  you  would  have  the  number  of  men  clouh'e,  triple,  or  quad- 
ruple, &c.  as  many  in  rmlc  as  in  file,  extrad:  the  fquare  root  of 
£,  f,|,  &c.  of  the  given  number  of  men,  and  that  will  be  the 
number  of  men  in  file,  which  double,  triple  qadruple,  &c.  and  the 
product  will  be  the  number  in  rank. 

f  The  above  rule  will  be  found  ufeful  in  planting  •trees,  having 
the  diftance  of  ground  between  each  given, 


SQUARE  ROOT.  173 

7~yards  ;  how  many  trees  mu  1:  it  be  in  length,  and  how 
many  in  breadth  ;  and  how  many  fquare  yards  of  ground 
do  they  ftand  on  ? 

To  refolve  any  queiUonof  this  nature,  fay,  as  the  ratio 
in  length  :  is  to  the  ratio  in  breadth  ::  foisthe  number 
of  trees  :  to  a  fourth  number  ;  whofe  fquare  root  is  the 
number  in  breadth  ;  And  as  the  ratio  in  breadth  :  is  to 
the  ratio  in  length  ::  fo  is  the  number  of  trees  to  a  fourth,- 
whofe  root  is  the  number  in  length. 

As  3  :  2  ::  600  :  400.  And  ^400=2 o=numbsr  in 
breadth. 

As.  2  :  3  ::  600  :  900.  And  -v/9°°=3°=:ium^er^n 
length. 

As  i  :  7  ::  30 — I  :  203.  And  as  i  :  7  ::  20 — *  : 
133.  And  203X133=26999  fquare  yards,  Ans. 

PROS  VII.  Admit  a  leaden  pipe  |  inch  diameter  will 
fill  a  ciilern  in  3  hours ;  I  demand  the  diameter  of  ano- 
ther pipe,  which  will  fill  the  fame  cittern  in  i  hour. 

RULE  As  the  given  time  is  to  the  fquare  of  the  giv- 
en diameter,  fo  is  the  required  time  to  the  fquare  of  the 
required  diameter. 

£=•75  :  and  '75><'75- 5625-  Then,  as  3!!.  :  5625  :: 
ih.  :  16875  inverfely,  and  ^i-6875^1-3  inch  near^ 
ly,  Anfwer.  . 

PROB.  V1IE.  If  a  pipe,  whofe  diameter  is  1-5  inch, 
fill  a  ciliern  in  5  hours,  in  what  time  will  a  pipe,  whofe 
diameter  is  3-5  inches,  fill  the  fame  ? 

1-5X1-5=2  2j  ;  and  3  5X3-5=1 2-25.  Then,  as  2-25  : 
5  ::  12*25  :  '91 84-  hou^  inversely, =55  minutes  5  fec- 
onds,  Aniwer,  . 

PROB.  IX.     If  a  pipe  6  Inches  bore,  will  be  4  hmrs  in 
running  off  a  certain  quantity  of  water  ;  in  what  time  will 
3  pipes,  each  4  inches  bore,  oe  ia  difcharging  doujle   the- 
quantity  ? 

6x6=36  4X4=16,  and  16X3=48.  Then,  as  36  :  4h. 
::.  48  :  3h  inverfely,  and  as  i\v.  :  3h.  ::  2W\  :  6.1. 
Anfwer. 

PROB.  X  Given  the  diameter  of  a  circle  to  make  an- 
other circ'e,  which  fhdli  be  2,  3,  4,  5cc«  tim?s  greater  or  lefs. 
than  the  given  circle. 

P    2 


174  SQUARE  ROOT. 

RULE. — Square  the  given  diameter,  and  if  the  required 
circle  be  greater,  multiply  the  fquare  of  the  diameter  by 
the  given  proportion,  and  the  root  of  the  product  will  be 
the  required  diameter  ; — But  if  the  required  circle  be  lefs  ; 
divide  the  fquare  of  the  diameter  by  the  given  proportion, 
and  the  root  of  the  quotient  will  be  the  diameter  required. 

There  is  a  circle,  whofe  diameter  is  4  inches  :  I  de- 
mand the  diameter  of  a  circle  3  times  as  large. 

4X4=16  ;  and  16X3=48  ;  and  ^48=6-928+  inches, 
Anfwer. 

PROS.  XI.  To  find  the  diameter  of  a  circle  equal  in 
area  to  an  ellipfis  (or  oval)  whofe  tranfverfe  and  conjugate 
diameters  are  given.* 

RULE. — Multiply  the  two  diameters  of  the  fcJlipfis  to- 
gether  ;  and  the  Square  Root  of  that  product  will  be  the 
diameter  of  a  circle  equal  to  the  ellipfis. 

Let  the  tranfverfe  diameter  of  an  ellipfis  be  48,  and  the 

conjugate  36:  What  is  the  diameter  of  an   equal   circle  ? 

48X36=1728,  and  v'1 728=4 j -5694-  the  Anfwer. 

NOTE.  The  fquare  of  the  hypothenufe,  or  the  longefl 
fide  of  a  right  angled  triangle,  (by  47th  B.  i,  Euc.) 
*s  equal  to  the  fum  of  the  fquares  of  the  other  two  fides  ; 
and  confequently  the  difference  of  the  fquares  of  the  hy- 
pothenufe and  either  ot  the  other  fides  is  the  fquare  of 
the  remaining  fide. 

PROB.  XII.  A  line  36  yards  long  will  exactly  reach 
from  the  top  of  a  fort  to  the  oppofite  bank  of  a  river, 
known  to  be  24  yards  broad.  The  height  of  the  wall  is 
required  ? 

36X36=1296  ;  and  24X24=576  Then,  1296 — 5/6= 
720,  and  \/ 7  20=  2 6  834-  yards,  the  Anfwer. 

PROB.  XIII.  The  height  of  a  tree,  growing  in  the 
centie  of  a  circular  iflund  44  feet  in  diameter,  is  75  feet, 
and  a  line  ftretched  from  the  top  of  it  over  to  the  hither 
edge  of  the  water  is  256  feet  What  is  the  breadth  of 
the  (bream,  provided  the  land  on  each  fide  of  the  water 
be  level  ? 

*  The  tranfverfe  and  conjugate  are  the  longeft  and  fhorteft  cliim- 
fters  of  an  eliipfis ;  they  pafs  through  the  centre,  and  crofs  each  otfc? 
T  at  right  angles. 


CUBE  ROOT.  175 

256x2 ^6=65536;  and  7-X75=>-)25  :  Then  65536 — 
562^=59911,  and  ^5991 1=244  76+  ar.d  244-76— 4,/=; 
222/7,6  fffify  Anfwer. 

PiK^r^V.  Suppofe  a  ladder  60  feet  long  be  fo 
plantejfcas  to  reach  a  window  37  feet  from  the  ground, 
on  one  nde  of  the  ftreet,  and  without  moving  it  at  the 
foot,  will  reach  a  window  23  feet  high  on  the  other 
fide  ;  I  demand  the  breadth  of  the  ftreet. 
»  60X60=3600  37X37='t3^9-  23X23=529-  Then, 
3600 — 1369=2231,  and  v/223  I=47  23+>  and  3600 — 
529=3071,  and  v/3°71=55'4i+>  then  47*23+55'4I:= 
1 02  64  feet,  Ans. 

PXOB.  XV.  Two  fhlps  fail  from  the  Ame  port  ;  one 
goes  due  north  45  leagues,  and  the  other  due  weft  76 
leagues  :  How  far  are  they  afunder  ? 

45X45=2015.  76X76=57^6.  The*),  5776+2025=7801 
and  -y/78oi  =88-32  leagues,  the  Anfwer. 

EXTRACTION  OF  THE  CUBE  ROOT. 

A  Cube  is  any  number  multiplied  by  its  fquare  To 
extract  the  cube  root,  is  to  rind  a  number  which  being 
multiplied  into  its  fquire,  (hall  produce  the  given  number 

RULE. 

1.  Separate  the  given  number  into  periods  of  three 
figures  each,  by  putting  a  point  over  the  unit    figure  and 
every  third  figure  beyond  the  place  of    units. 

2.  Find  the  greateft  cube  in  the  left  hand  period,    and 
put  its  root  in  the  quotient 

3.  Subtract  the  cube  thus  found,  from  the  faid  period, 
and  to  the  remainder  bring  down  the  next  period,  and  call 
this  the  dividend. 

4.  Multiply  the  fquare  of  the  quotient  by    300,  calling 
it  the  triple  fquare,  and  the  quotient  by    30   calling  it  ths 
triple  quotient,  and  the  fum  of  thefe  call  the  divifor. 

5.  Seek  how  often  the  divifor  may  be  had  in  the   divi- 
dend, and  place  ths  refuitin  the  quotient 

6.  Multiply  the  triple  fquare  by  the   lift   quotient  fig- 
ure, and  write  the  product  u:;de:  th'   dividend  ;    multiply 
the  fquare  of  the  lad  quotient  figure  by  the  triple  quotient, 


CUBE  ROOT. 

and  place  this  product  under  the  laft  ;  under  all,  fetthe 
cube  (if  the  lail  quotient  figure  and  call  their  fum  the  iub- 
trahend.  ,Jr*  , 

7.  Subtract  the  fubtrahend  from  the  dividttw^tnd  to 
the  remainder  bring  down  the  T.exr  period  for  a  n^fr  div- 
idend, with  which  proceed  as  before,  and  fo  oivfill  the 
whole  be  tiniihed. 

NOTE.  The  fame  rule  muftbe  obferved  for  continuing 
the  operation  and  pointing  for  decimals,  as  in  the  fquare  . 
root. 

EXAMPLES. 

i.  Required  the  cube  root  of  436036824287. 

436036824287(7583  the  root 
343 

jft  Divifor=  1 49 10)93036—1  ft  Dividend« 


7 88 7 5=1  ft  Subtrahend, 
2d  Divifor=i 689750)  141 6 1 824=zd  Dividend. . 


13500000- 
144000 
512 

1 364.45 1 2=d  Subtrahend. 


3d  Divifor^!7239i94o)p73i2287=3d  Dividend. 

517107600 
204660  . 
27 


5 1 73 1 2  28  7=3  d  Subtrahend. 


CUBE  ROOT.  177 

The  Method  of  Operation. 

7X7X300  =     14700  =   ift  Triple  fquare. 
7X30         =         210  =   i  ft  Triple  quotient, 

14910  =   i  ft  Divifor. 

14700X5  =  73500 
5X5x210  ~  5250 
5X5X5  =  125 

78*75  =  i  ft- Subtrahend. 


75X75X300  =     16*7500  =  2 d  Triple  fquare. 
75X30  2Z50  =  2d  Triple  quotient 

1689750  =  2d  Divifor. 

1687500X8  =  13500000 
2250X8X8  =  144000 
8x8x8  =  512 

13644512  =  2d  Subtrahend. 


758X758X300  =172369200  =  3d  Triple  fquare. 
758X30  =         22740  =  3d  Triple  quotient, 

172391940  =  3d  Divifor. 

172369200X3  =517107600 
22740X3X3  =  204660 
3X3X3  =  27 


51731/287  s=  3d  Subtrahend. 

2.  What  is  the  cube  root  of  34965783  ?       Ans.  327* 

3.  What  is  the  cube  root  of  84-6045  19  ?      Aris.  4'39» 

4.  Whit  is  the  cube  root  of  -008649  ?      Ans.  '2052+. 

5.  What  is  the  cube  root  of  -f£  ?  Ans.  I. 


178  CUBE  ROOT. 

TQ  find  the  true  denominator ',  to  bs  placed  under  the  remainder '9 
after  the  operation   is  finfflcd. 

In  the  extraction  of  the  cube  root,  the  quotient  is  faid 
to  be  fquared  and  tripled  for  a  new  divilbr  :  But  is  not 
really  fo,  till  the  triple  number  of  the  quotient  be  added 
to  it ;  therefore,  when  the  operation  is  finifhed,  ic  is  but 
Squaring  the  quotient,  or  root,  then  multiplying  it  by  3, 
and  to  that  number  adding  the  triple  number  of  the  root, 
when  it  will  become  the  divifor,  or  true  denominator  to 
its  own  fraction,  which  fraction  muit  be.  annexed  to  the 
quotient  to  complete  the  root. 

Suppofe  the  root  to  be  12,  when  fqnared  it  will  be  144, 
and  multiplied  by  3  it  makes  432  to  which  add  36,  the 
triple  number  of  the  root,  and  it  produces  468  for  a  denom- 
inator. 

BY    APPROXIMATION. 

1.  A/Tume  a  root,  whofe  cube  will  probably  be  near  the 
given  number  ;    then  multiply  the  Iquare  of  the  aflumed 
rout  by  3,  and  divide  the  rtioivend    by  this   product  ;  to 
this  quotient  add  f  of  the  jfTumed  root,  and  the  ium  will 
be  the  true  root,  or  an  approximation  to  it. 

2.  For  each  iiicceeding  operation  let  the   laft  approxi- 
mated root  be  the  affuuied  root,  and   proceeding   in   this 
manner,  the  root  may  be  extracted    to  any  ailigned  ex- 
ac'tneis. 

1.  What  is  the  cube  root  of  7  ? 

Let  the  affumed  root  be  2.  Then  2X2X3=12  the  di- 
vifor. 

I2)7'°(  583  5  to  this  add  f-  of  2=1-333,  &.C..  that  is, 
•5  83+1  '3 3  3=  i  916  approximated  root. 

Now  ailume  1916  for  the  root.     Then,  by  the   fecond 

7 

procefs,   the  root  is    x~^f~7"Q+|X  1*9 16=1*9 126,  £c. 

2.  What  is  the  cube  root  of  9  j?     Let  2  be  the  affum- 
ed  root  as  before.     Then,  T9g-t-|X2=2  c8  the  approxuaat- 

9  

ed  root.    Now  ailume  2  08.     Then  ^'xTc^'2  +  *X2  °^ 

=$.,08008,  &c. 


CUBE  ROOT.  i79 

3.  What  is  the  cube  root  of  2^2  ?     Let  6  he  the  afTum- 
ed  root.     Then  6x6x3=108)282(2  61  i,  £c   and  2*61  i-l-f 
of  6=6'6f  r  appsox'iTMted  root.     N«"\v  nffurae  6  61  r,  and 
it  will  be  6'6t  1x6     1 1X3=131-116)282(2  1^07,   &c.    and 
2*1507+1  of  6  61  1=6-558  a  farther  approximated  root. 

4.  What  is  the  cube  root  of  1728  ?     Here  the  a/Turned 
root  is  io-     Then,  !oXioX<=3oo)i  7*8(5 -76,  and  5  y6+| 
of    10— 12-426         Now    aflame    12-4^6,     then    i242ox 
12  4*6x3=426-2  6428)1728(3  7  ?2,     and     3-732  -f-J    of 
l^  426=  2  014  a  farther  approximated  root,  and  fo  on. 

APPLICATION  AND  USE  OF  THE  CUBE 
ROOT. 

i.  To  find  tiuo  mean  proportionals  between  any  two 
given  numbers. 

RULE. 

1.  Divide  \\izgreater  by  the  kfi>  and  extra<5l  the  cube 
root  of  ;he  quotient. 

2.  Multiply  th^  root,  fo  found,  by  the  leaft  of  the  given 
numbers,  aisdtht  product  will  be  the  leaft. 

3  Multiply  this  product  by  the  fame  root,  and  it  will 
give  the  greateft 

EXAMPLES. 

1.  What  are  the  two  mean  proportionals   between  6 
and  750  ? 

3 

•7^0^-6=12^,  and  \/  125=9.  Then  5X6~3o=!eaft, 
and  30x^  =  1 5o=greateit  Ans.  30  and  150. 

Proof.      As  6  :   30  ::    150  :   750. 

2.  What  are  the  two  mean  proportionals  between  56 
and  12096?  Ans    336  and  2016. 

NOTE  The  folid  contents  of  fimilai  figures  are  in 
proportion  to  each  other,  as  the  cubes  of  their  fimilar 
fides  or  diameters 

3.  If  a  bullet  6  inches  diameter  weigh  32lb.  ;    what 
will  a  bullet  of  the  fame  metal   weigh  whofe   diameter  is 
3  inches  ? 

6x6x6=216.      3x3x3=27. 

As  216  :  32ib.  ::  27  ;  4lb,  Ans. 


i8o  CUBE  ROOT. 

4.  If  a  globe  of  fiiver  of  3  inches  diameter  be  worth 
£45,  what  is  the  value  of  another    globe,  of  a  foot  di- 
ameter ?  3X3X3=27       i2X;2Xf2=i728. 

As  27  :  45  ::   1728  :  £2880  Ans. 

2d.  The  fide  of  a  cube  being  given,  to  find  the  fide  oi 
that  cube  which  (hall  be  double,  triple,  &c.  in  quantity  to 
the  given  cube. 

RULE.  Cube  your  given  fide,  and  multiply  it  by  the 
given  proportion  between  the  given  and  required  cube,  and 
the  cube  root  of  the  product  will  be  the  fide  fought. 

5.  If  a  cube  of  fiiver,  whofe  fide  154  inches,  be  worth 
£  50, 1  demand  the  fide  of  a  cube  of  the  like  fiiver,    whofe 
value  (hall  be  4  times  as  much  ? 

4X4X4=64,  and  64X1=256.     v'256=6*349-Hnches,  Ans. 

6.  There  is  a  cubical  vefTe*,    whofe    fide  is  2   feet  ;  I 
demand  the  fide  of  a  veffel,  which  thall  contain  three  times 
as  much  ?  3 

2x2X2=8,  and  8X3=24.      v'24=2*884=2ft.   loj-in.  Ans. 

7.*  The  diameter  of  a  buihel  meafure  being  18^  inch- 
es, at»d  the  height  8  inches,  I  demand  the  fide  of  a  cuuick 
box,  which  (hall  contain  that  quantity  ? 

Ans    12*907+ inches. 

8.  Suppofe  a  fhip  of  500  tons  has  89  feet  keel,  36  teet 
beam,  and  16  feet  deep  in  the  hold  ;  what  are  the  di~ 
menfions  of  a  (hip  of  200  tons,  of  the  fame  mould  and 
and  fh-ipe  ? 

89X89X89=704969=cubed  keel. 

As  500  :  200  ::  704969  :  281987*6  cube  of  the  re- 
quired keel. 

,v/28i987'6=65'57  feet  the  required  keel. 

As  89  :  65  57   ::  36  :   26-522=26^  feet  beam,  nearly. 

As  ^9  :  65-57  ::  16  :  11-7  feet,  depth  of  the  hold, 
nearly. 

3d  From  the  proof  of  any  cable  to  find 'the  ftrength 
of  any  orher. 

RULE.  The  ftrength  of  cables,  and  confequently  the 
weights  of  their  anchors,  are  as  die  cubes  of  their  pe- 
ripheries. 

*  Multiply  the  fqime  of  the  diameter  by  7854,  and  the  produ& 
by  the  height ;  the  cube  root  of  the  Jaft  product  is  the  anfwer, 


EXTRACTION  OF  ROOTS.  18 1 

9.  if  a  cable  12  inches  about  require  an  anchor  of  18 
cwt.  of  what  weight  nmft  an  anchor  be  for  a  15  inch  ca- 
ble ?  Cwt.  Cwt. 

As  12X12X12  :   18  ::   15X15X15  :  35-15625,  Ans. 

10.  If  a  1 5  inch  cable  require  an  anchor  35T5625cwt. 
what   muft  the   circumference  of  a  cable  be,   for  an  an- 
chor of  i  Scwt.  ?  3 

As  35*15625  :  15X15X15  ;:  18  :  1728,  and  ^1728 
=  12,  Anfwer, 

EXTRACTIOSfibFTHE  BlgUADRATE  ROOT. 
RULE. 

Extract  the  fquare  root  of  the  refolvend,  and  then  the 
fquare  root  of  that  root,  and  you  will  have  the  bi qua- 
drate root. 

What  is  the  biquadrate  root  of  20736  ? 

207 36(  144  i44(  12  root  required. 

i  i 

22)44 
44 


1136 


A  GENERAL  RULE* 

FOR    EXTRACTING    THE    ROOTS    OF    ALL    POWERS. 

i.  Prepare  the  given  number  for  extraction,  by  point- 
ing off  from  the  unit's  place,  as  the  required  root  direfts. 

*  The  extracting  of  roots  of  very  high  powers  will,  by  this  rule, 
be  a  tedious  operation  :-^-Phe  following  method,  when  practicable, 
will  be  much  more  convenient. 

When  the  index  of  the  power,  whofe  root  is  to  be  extracted,  is  a 
composite  number,  take  any  two  or  more  indices,  whofe  product  is 
equal  to  the  given  index,  and  extract  out  of  the  given  number  a 
root  anfweriog  to  another  of  the  indices,  and  fo  on  to  the  laft 

Thus,  tiie  fourth  roo!=fquare  root  of  the  fquire  root  :  the  fixth 
root=fquare  root  of  the  cube  root  ;  —  the  eighth  root  =  :q>iare  root 
of  tin-  fourth  root  ;  —  the  ninth  rooi=the  cube  root  of  the  cube  root  ; 
—the  tcmh  ro!)r=fquAit  root  ot  the  fifth  root  ^*—  the  twelfth  root= 
cube  root  of  the  fourth,  &c, 


i§2  EXTRACTION  OF  ROOTS. 

2.  Find  the  firft  figure  of  the  root  by  trial,  or  by  in- 
fpeftion  into  the  Table  of  Powers,  and  fubtracl:  its  power 
from  the  left  hand  period. 

3.  To  the  remainder  brine^  down  the  firft:  figure  in  the 
next  period,  and  call  it  the  dividend. 

4.  Involve  the  root  to  the  next  infcriour  power  to  that 
which  is  given,  and  multiply  it  by  the  number,    denoting 
the  given  power,  for  a  divijbr* 

5  Find  how  many  times  the  divifor  may  be  had  in 
the  dividend,  and  the  quotient  will  be  another  figure  of 
the  root 

6.  Involve  the  whole   root  to  the  given   power,  and 
fub tract  it  from  the  given  number  as  before. 

7.  Bring  down  the  firft  figure  of   the  next  period  to 
the  remainder  for  a  new  dividend,  to  which  find  a  new 
divifor,  as  before,  and,  in  like  manner,   proceed   till    the 
whole  be  finifhed. 

EXAMPLES. 
I.  What  is  the  cube  root  of  20346417  ? 

20346417(273 
23         =    8  =   i  ft  Subtrahend. 

=.  Dividend. 

273=      19683       =2d  Subtrahend. 
27^x3=2 1 87)6634    ==  2d  Dividend. 

3        =     20346417=  3d  Subtrahend. 


2X2X2=8  root  of  the  ift  period,  or  ift  fubtrahend. 

2X2=4(=next  inferi our  power)  and, 

4X3  (the  index  of  the  given  power}— 12=1  ft  divifor. 

2 7X2 7X2 7=1 9683  =  2d  fubtrahend. 

27X27=729  (=next  inferiour  power)  and, 

729X3   (=index  of  the  given   power)=2i87=2d  div. 

273X273X273=273464 1 7=3d  fubtrahend. 

2.  What  is  the  biquadrate  root  of  348279^8976  ? 

Ans.  43 i  9+ . 

3.  Extract    the    fquare    cubed,    or     fixth    root    of 
1178420166015625.  Ans.  325. 


PROPORTION.  183 


OF  PROPORTION  IN  GENERAL. 


NL7M3EP.S  are  compared  together,  to  difcover  the 
relations  they  have  to  each  other. 

There  muft  be  two  numbers  to  form  a  comparison  : 
The  number,  which  is  compared,  being  written  firft,  is 
called  the  antecedent  ;  and  that,  to  which  it  is  compared, 
the  confequent. 

Numbers  are  compared  with  each  other  two  different 
ways  :  The  one  companion  confiders  the  difference  of  the 
two  numbers,  and  is  called  Arithmetical  Delation,  the 
difference  being  fometimes  named  the  Arithmeti:al  Ratio; 
and  the  other  confiders  their  quotient^  which  is  termed 
Geometrical  Relation,  and  the  quotient,  the  Geometrical 
Ratio.  Thus,  of  the  numbers  j  2  and  4,  the  difference, 
or  Arithmetical  Ratio,  is  12 — 4=8  ;  and  the  Geometric 

12 

cal  ratio  is  —  =3.* 

If  two  or  more  couplets  of  numbers  have  equal  ratios,  cr 
differences,  the  equality  is  termed  proportion  ;  and  their 
terms,  fimilirly  pofited,  that  is,  either  all  the  greater,  or 
all  the  lefs  taken  as  antecedents,  and  the  reft  as  confequents, 
are  called  proportionals.  So,  the  t\vo  couplets,  2,  4,  and 
6,  8,  taken  thus,  2,  4,  6,  8,  or  thus,  4,  2,  8,  6,  arc  arith- 
metical proportionals  ;  and  the  two  couplets,  2,  4,  and 
8,  1 6,  taken  thus,  2,  4,  8,  16,  or  thus,  4,  2,  16,  8,  are 
geometrical  proportionals. f 

Proportion  is  diitinguilhed  into  continued  and  difcon- 
tinned.  If,  of  feveral  couplets  of  proportionals,  written 
down  in  a  feries,  the  difference  or  ratio  of  each  confe- 

*  RATIOS  are,  here,  always  confidered.as  the  refult  of  the  greater 
term  of  companion  diminilhed,  or  divided  by  the  tefs  ;  not  regard- 
ing which  of  them  be  the  antecedent. 

f  Four  numbers  are  faidto  be  reciprocally  or  invtrfely  proportional, 
when  the  fourth  is  Ic-fs  than  the  fecond  hy  a*  many  times  as  the  third 
is  greater  than  thtlirft,  or  when  the  firft  is  to  the  third,  as  the  fourth 
to  the  fecond,  and  vice  vsr/a.  Thus,  a,  9,  6,  aad  3  are  reciprocal 
proportionals. 

NOTE.  It  is  common  to  read  the  geometrical  2:4  '•>  8  :  16, 
thus,  a  is  to  4  as  8  to  16  ;  or,  aa  a  to  4  fo  is  8  to  16. 


i84        ARITHMETICAL  PROPORTION. 

quent,  and  the  antecedent  of  the.  next  following  couplet > 
be  the  fame  as  the  common  difference  or  ratio  of  the 
couplets,  the  proportion  is  faid  to  be  continued,  and  the 
numbers  themfelves,  a  feries  of  continued  arithmetical  or 
geometrical  proportionals.  So  2,  4,  6,  8,  form  an  arith- 
metical progreffion  ;  for  4 — 2=6 — 4=8 — 6  ;  and  2,  >4t 
8,  1 6,  a  geometrical  progreffion  ;  for  j=-J=!-y  =2. 

But,  ij  the  difference  or  ratio  of  the  confequent  of  one 
couplet,  and  the  antecedent  of  the  next  couplet  be  not  the 
fame  as  the  common  difference  or  ratio  of  the  couplets,  the 
proportion  is  faid  to  be  difcontinued.  So  4,  2,  8,  6,  are  in 
difcontinued  arithmetical  proportion  ;  for  4 — 2=r8 — 6zr2 
—common  difference  of  the  couplets,  8 — 2=6=difference 
of  the  ccnfequent  of  one  couplet  and  the  antecedent  of 
the  next  ;  alib,  4,  2,  16,  8  are  in  difcontinued  geometric- 

4       16 

al  proportion  ;  for  —  = —  =  2  =   common  ratio  of  the 
2        8 

couplets,  and  —  =8=ratio  of  the  confequent  of  one 
couplet  and  the  antecedent  of  the  next. 


ARITHMETICAL   PROPORTION. 

THEOREM  i.  If  any  four  quantities,  2,  4,  6,  8,  be  in 
Arithmetical  Proportion,  the  fum  of  the  two  means  is 
equal  to  the  fum  of  the  two  extremes. 

And  if  any  three  quantities,  2,  4,  6,  be  in  Arithmetical 
Proportion,  ihe  double  of  the  mean  is  equal  to  the  fum 
of  the  extremes. 

THEOREM  2.  In  any  continued  arithmetical  propor- 
tion, i,  3,  5,  7,  9,  ii,  the  fum  of  the  two  extremes,  and 
and  that  of  every  other  two  terms,  equally  diftant  from 
them,  are  equal :  Thus,  i4-/ 1=3+9=5+7. 

When  the  number  of  terms  is  odd,  as  in  the  proportion, 
3,  8,  13,  18,  23,  then,  the  fum  of  the  two  extremes,  being 
double  to  the  mean,  or  middle  term,  the  fum  of  any  other 
two  terms  equally  remote  from  the  extremes  muft  likewife 
be  double  to  the  mean, 


ARITHME  PIC  AL  PROGRESSION.       1 85 

THEORETJ  t.  In  any  continued  arithmetical  propor- 
tion 4,  4+  f-t  4+4*  4*6»  4+8,  &c.  the  lail  or  greatetl  term 
is  equal  to  the  firft  or  lead  more  the  common  difference 
of  the  terms  drawn  into  the  number  of  all  the  terms  af- 
ter the  firft,  or  into  the  whole  number  of  Uie  terms  lefs 
one. 

THEOREM  4.  The  fum  of  any  rank,  or  ferfes,  of  quan- 
tities in  continued  arithmetical  proportion,  i,  3,  5,  7,  9, 
1 1,  is  equal  to  the  fum  of  the  two  extremes  multiplied  into 
half  the  number  of  terms,* 


ARITHMETICAL  PROGRESSION. 

ANY  rank  of  numbers,  more  than  two,  increafmg  by  a 
common  excefs,  or  decreafmg  by  a  common  difference, 
is  faid  to  be  in  Arithmetical  Progreflion. 

If  the  fucceeding  terms  of  a  progreffion  exceed  each 
other,  it  is  called  an  afcending  feries  or  progrefijon  ;  if  the 
contrary,  a  defcending  itries. 

*  The  fame  thing  alfo  holds,  when  the  number  of  terms  is 
odd,  as  in  the  feries,  4,  8,  12,  16,  20  ;  for  then  the  mean,  or 
riiddle  term,  being  equal  to  half  the  fum  of  any  two  terms  equally 
difUnt  from  it  ou  contrary  fides  it  >»  obvious  that  the  value  of.'  the 
whole  feries  is  the  fame  as  if  every  ttrm  thereof  were  equal  to  the 
mean,  and  therefore  is  equal  to  the  mean  (or  half  the  I'um  of  the 
two  extremes)  multiplied  by  the  whole  number  of  terms  ;  or  to  the 
fum  of  the  extremes  multiplied  by  half  the  number  of  terms. 

The  fum  of  any  number  of  terms  of  the  arithmetical  feries  of  odd 
numbers,  x,  3,  5,  7,  9,  &c.  is  equal  to  the  fquare  of  that  number. 

For,  o-t- 1  or  the  fum  of  i  term=  i  *  or    i 

1+3  or  the  fum  of  2  terms=2*  or    4 

4+5   or  the  fum  of  3  terms=32  or    9 

9+7  or  the  fum  of  4  term ^=4*  or  16 

1 6+9  or  the  fum  of  5  terms=52  or  25,  &c. 

EXAMPLE. 

The  firft  term,  the  ratio,  and  number  of  terms  given,  to  find  the 
"am  of  the  feries. 

A  Gentleman  travelled  19  days  ;  the  firft  day  lie  went  but  i  mile, 
and  increafed  every  day's  travel  a  miles  :  How  far  did  he  travel  ? 

29X^9=841  mJes  the  anfv/er. 


iS6      ARITHMETICAL  PROGRESSION. 


s 


"o,  2,  4,  6,     8,   10,  &c.  is  an  afcending  arithmetic- 
s     .  al  feries. 

2,  4,  8,   1 6,  32,  &c.  is  an  afcending  geometric- 
al feries. 

10,     8,  6,  4,  2,  o,  &c.  is    a   defcending    arith- 
A    ,   .  metical  feries. 

32,   16,  8,  4,  2,   i,  &c.  is  a  defcending  geomet- 
_  rical  feries. 

The  numbers,  which  form  the  feries,  are  called  the 
terms  of  the  progreffion. 

NOTE.  The  firft  and  laft  terms  of  a  progreffion  are  call- 
ed the  extremes,  and  the  other  terms  the  means. 

Any  three  of  the  five  following  things  being  given,  the 
other  two  may  be  eafily  found. 

1.  The  firft  term. 

2.  The  la(t  term. 

3.  The  number  of  terms. 

4.  The  common  difference. 

5.  The  fum  of  all  the  terms. 

PROBLEM    I. 

The  firft  tern?)  the  laft  term,    and  the  number  of  terms  being 
giveny  tojind  the  common  difference. 

RULE. 

Divide  the  difference  of  the  extrtmes  by  the  number 
of  terms  lefs  i  and  the  quotient  will  be  the  common  cttf- 
rerence  fought. 

EXAMPLES. 

i.  The  extremes  are  3  and  39,  and  the  number  of 
terms  is  19  :  What  is  the  common  difference  ? 


Divide  by  the  number  of  terms — i='9 — 1=1^)36(2  Ans. 

18 

39—3 
Or, =2 

19—1 

2.  A  man  had  to  fons,  whofe  feveral  ages  differed 
alike  ;  the  youngeft  was  3  years  old,  and  the  eldeft  48  ; 
What  was  the  common  difference  of  their  ages  ? 

4*-3 

=5 

ic — i 


ARITHMETICAL  PROGRESSION.       187 

3.  A  man  is  to  travel  from  Bofton  to  a  certain  place 
in  9  days,  and  to  go  but  5  miles  the  firft  day,  increa  :ng 
each  day  by  an  equal  excefs,  fo  that  the  laft  day's  journey- 
may  be  37  miles  ;  required  the  daily  increafe. 

37—5 
—  —  —  =4 


PROBLEM   II. 

'The  firft  term,  the  laft  term,  and  the  number  cf  terms  bang 
given,  to  find  the  fum  of  all  the  terms. 

RULE.  Multiply  the  fum  of  the  extremes  by  the  num- 
ber of  terms,  and  half  the  product  will  be  the  anfwer. 

EXAMPLES. 

i.  The  extremes  of  an  arithmetical  feries  are  $  and  39, 
and  the  number  of  terms  1  9  j  required  the  fum  of  the 
feries  ? 


j. 


Number  of  terms  =1X19 

373 
42 

2)798 

39+3X19  

Or, =,399  399  Ans. 

2 

2.  It  is  required  to  find  how  many  ftrokes   the  ham- 
mer of  a  clock  would  (Irike  in  a  week,  or  168  hours,  pro- 
vided it  increafed  i  at  each  hour  ? 

168+  i  Xi 68 

— — — • =  14196  Anfwer* 

2 

3.  Suppofc  a  number  of  ftones  were  laid  a  yard  dif- 
tant  from  each  other  for  the  fpace  of  a  mile,  and  the  firft 
a  yajd  from  a  bafkst  j  what  length  of  ground  will  that 


1 88       ARITHMETICAL  PROGRESSION. 

man  travel  over  who  gathers  them  up  flngly,  returning 
with  them  one  by  one  to  the  bafket  ? 

3520+2X1760 

,,.„ r-  =  3099360  yards=i76«  miles,  Ans, 

2 

N  B.  In  this  queftion,  there  being  1760  yards  in  a 
mile,  and  the  man  returning  with  each  Hone  to  the  baiket, 
his  travel  will  be  doubled  ;  therefore  the  firlt  term  will 
be  2  and  the  laft  1760X2,  and  the  number  of  terms  1760. 

4.  \   man  bought  25  yards  of  linen  in    arithmetical 
progrefllon  ;  for  the  4th  yard  he  gave  i  2  cents,  and  for 
the  laft  yard  75  cents.     What  did  the  whole  amount  to,, 
and  what  did  it  average  per  yard  ? 

75—12 

•  =  3  the  common  difference,  by  which  the  firft 

22—1         term  is  found  to  be  3. 

75+3^5 

Then    — - —  =  90.  75C.  and  the  average  price  is  39 
2  cents  per  yard. 

5.  Required   the  fum  of  the   firft    1000   numbers  in 
their  natural  order  ?  — 

icoo-f  1X1000 
=500500  Ans. 

2 

PROBLEM    III. 

Given  the  extremes  and  the  common  difference^  to  find  the  num* 

ber  of  terms. 

RULE.     Divide  the  difference  of  the  extremes  by   the 
common  difference,  and  the  quotient  increafed  by  i  will 
be  the  number  of  terms  required. 
EXAMPLES. 

i.  The  extremes  are  3  and  39,  and  the  common  differ- 
ence 2  :  What  is  the  number  of  terms  ? 


Common  difference=2)36 
Quotient=i8 


Extremes. 


"19  Anfwer, 


ARITHMETICAL  PROGRESSION.       i&9 

39—3 
Or, +  i  =  19. 

2 

2.  A  man  going  a  journey  travelled  the  firft  day  7 
miles,  the  laft  day  5 1  miles,  and  each  day  increafed  his 
journey  by  4  miles  :  How  many  days  did  he  travel,  and 

how  far  ?  

51—7  51+7X12 

hi=i2  days,  and ———=348  miles,  Ans, 

4  2 

PROBLEM  IV. 
T/}ff  extremes  and  common  difference  given,  to  find,  tke  fum  of 

the  f cries, 

RULE.  Multiply  the  fum  of  the  extremes  by  their 
difference  increafed  by  the  common  difference,  and  the 
product  divided  by  twice  the  common  difference  will  give 
the  fum. 

EXAMPLES. 

I.  If  the  extremes  are  3  and  39,  and  the  common  dif- 
ference 2,  what  is  the  fum  of  the  ieries  ? 
39+3=42=fum  of  the  extremes. 
39 — 3=36=d:fference  of  the  extremes. 
36-t-2=38=diflrerence  of  the   extremes  increafed  by  the. 
common  difference.  42 

X38 

336 
126 

Twice  the  common  difference^)  1596 

399  Ans, 

39-^x39—3+2 

Or, =  39c, 

2X2 

2.  A  owes  B  a  certain  fum,  to  be  difcharged  in  a 
year,  by  paying  6d.  the  firft  ;  week,  1 8d.  the  fecond,  and 
thus  to  increafe  every  weekly  payment  by  a  (hilling,  till 
the  laft  payment  be  £2  i  is.  6d.  what  is  the  debt  ? 

* 
=£67  i2s.  Ans, 


ipo      ARITHMETICAL  PROGRESSION. 

PROBLEM   V. 
The  extremes  and  fum  cf  ?he  feries  given,  to  find  the  common 

difference. 

RULE.  Divide  the  produft  of  the  fur/i  and  difference 
of  the  extremes  by  the  difference  of  twice  the  fum  of 
the  feries  and  the  fum  of  the  extremes,  and  the  quotient 
will  be  the  common  difference. 

EXAMPLES. 

1.  Let  the  extremes  be  3  and  39,  and  the  fum  399; 
what  is  the  common  difference  ? 

Sum  of  the  extremes=39+3=     42 
Difference  of  the  extremes:=39 — 3=  X  36 

252 

Z26 

399X2 — 42=756)1512(2  Acs* 

394-2X39—3 

Or, =2. 

399X2—39+3 

2.  A  owes  B  67!.  i2s.  to  be  difcharged  in  a  year,  by 
weekly  payments  ;    the   firft   payment  to  be  6d.  and  the 
laft  2!    iis.  6d.  :  What  is  the  common  difference  of  the 
payments,  and  what  will  each  payment  be  ? 


=  is.  and  6d.-f  is.=is.  6d.=2d  pay- 


1352x2— 51  - 

ment,  is.  6d.+is.=2s.  6d.=3d  payment,  £c. 

PROBLEM    VI. 

The  extremes  and  fum  of  the  feries  given,  to  find  the  numler 

of  terms. 

RULE.     Twice  the  fum  of  the  feries,  divided  by  the  fum 
of  the  extremes,  will  give  the  number  of  terms. 

EXAMPLES. 

Let  the  extremes  be  3  and  39,  and  the  fum  of  the  feries 
399  ;  what  is  the  number  of  terms  ? 


GEOMETRICAL  PROPORTION.        i9i 

Sum  of  the  feries:=399 

x     2 

Sum  of  extremes=39-f-3=42)798(i9  Ans. 

42 


399X2  378 

Or,  --  =  19. 

59+3 

2.  A  owes  B  67!.  us.  to  be  paid  weekly  in  arithmet- 
ical  progrefllon,  the  firft  payment  to  be  6d.  and  the  laft 
515.   6d.  :    How  many  payments  will  there  be,  and  how 
long  will  he  be  in  difcharging  the  debt  ? 
1352X2 

-  —52  payments,  and  as  many  v/eeks,  Ans. 
51-54-5 


GEOMETRICAL  PROPORTION. 

THEOREM  I.  If  four  quantities,  2,  6,  4,  12,  be  in 
geometrical  proportion,  the  product  of  the  means,  6X4, 
will  be  equal  to  that  of  the  two  extremes,  2X1 2,  whether 
they  are  continued,  or  difcontinued  ;  and,  if  three  quan- 
tities, 2  4,  8,  the  fquare  of  the  mean  is  equal  to  the  prod- 
uct of  the  two  extremes. 

THEOREM  2  If  four  quantities,  2,  6,  4,  12,  are  fuch 
that  the  product  of  two  of  them  2X12,  is  equal  to  the 
product  of  the  other  two,  6X4,  then  are  thofe  quantities 
proportional. 

THEOREM  3.  If  four  quantities,  2,  6,  4,  1 2,  are  propor- 
tional, the  rectangle  of  the  means,  divided  by  either  extreme, 
will  give  the  other  extreme. 

THEOREM  4.  The  products  of  the  correfponding  terms 
of  two  geometrical  proportions  are  alfo  proportional. 

That  is,  if  2  :  6  ::  4  :  12,  and  2  :  4  ::  5  :  10, 
then  will  2X2  :  6X4  ::  4X5  :  10X12.* 

*  And  if  any  quantities  be  proportional,  their  fquares,  cubes,  &c, 
•will  likewife  be  proportional. 


292        GEOMETRICAL  PROGRESSION. 

THEOREM  5.  If  four  quantities,  2,  6,  4,  12,  are  di- 
rectly proportional, 

1.  Directly,  z     :     6     ::     4     :   12 

2.  Inverfely,  6     :     2     ::   12     :     4 

3.  Alternately,  2     :     4     ::     6     :    12 

4.  Compoundedly,        2  :   6+2     ::     4  :   12+4 
T,              S-   Dividedly,  2  :   6—2     ::     4  :   12—4 

-n>       6    Mixtly,  6+2  :  6—2  ::    12+4  :   12—4 

7.  By  Multiplication,      2r   :     6r    ::     4     :   12 
or    2X5   :  6X5  ::  4  :   12 

2  6 

8    By  Divifion,  —    :   —    ::     4     ;   12 

r  r 

or,  £:£•::     4     :     is 

Becaufe  the  product  of  the  means  in  each  cafe,  is  equal 
to  that  of  the  extremes,  and  th  erefore  the  quantities  are 
proportional  by  Theorem  2. 

THEOREM  6.  If  three  numbers,  2,  4,  8,  be  in  contin- 
ued proportion,  the  fquare  of  the  firft  will  be  to  that  of 
the  fecond,  as  the  firft  number  to  the  third  -3  that  is, 
2X2  :  4X4  ::  2  :  8. 

THEOREM  7.  In  any  continued  geometrical  propor- 
tion, (i,  3,  9,  27,  8 1,  &c  )  the  product  of  the  two  ex- 
tremes, and  that  of  every  other  two  terms,  equally  diftant 
from  them,  are  equal. 

THEOREM  8.  The  fum  of  any  number  of  quantities,  in 
continued  geometrical  proportion,  is  equal  to  the  difference 
of  the  rectangle  of  the  fecond  and  laft  terms,  and  the 
fquare  of  the  firft  divided  by  the  difference  of  the  firft 
and  fecond  terms. 


GEOMETRICAL   PROGRESSION. 

A  GEOMETRICAL  Progreflion  is  when  a  Rank,  or 
Series,  of  numbers,  increafes  or  decreafes  by  the  continual 
multiplication,  ©r  divifion,  of  fome  equal  number. 

Pl<O3LEM     1. 

Given  one   of  the  extremes,  the  ratio*,  and  the  number  of  the 
terms  of  a  geometrical   feries,  to  find  the  other  extreme. 
RULE.     Multiply,  or  divide   (as  the  c^fe  may  require) 

the  given  extreme  by  fuch  power  of  the  ratio,  whofe  ex- 


GEOMETRICAL  PROGRESSION.        193 

ponent*  Is  equal  to  the  number  of  terms   lefs    i>  and  the 
produdl,  or  quotient,  will  be  the  other  extreme. 

EXAMPLES. 

i.  If  the  firft  term  be  4,  the  ratio  4,  and  the  number 
of  terms  9.  what  is  the  lait  term  ? 

*  As  the  la/I  term*  or  any  term  near  thelaft,  is  very  tedious  to  be 
found,  by  contmua  multipl-c  t  ion,  it  will,  often  he  neccllary,  in  or- 
der to  ascertain  them,  to  have  a  fcrics  of  numbers  in  arithmetical 
proportion,  called  indices  or  exponents^  beginning  with  a  cypher,  or  an 
unit,  whofe  common  diffeienceis  i. 

When  the  firjl  term  of  the  feries  and  (he  ratio  are  equal,  the  indices 
muft  begin  with  an  unit)  and,  in  this  cafe,  the  produift  of  any  t\vo 
terms  is  equ*l  to  that  term,  fignificd  by  the  fum  of  their  indices. 

„,,         (I,  2,  3,     4,     5,     6.   &c.   indices  or  arithmetical  feries. 

US  t  2>  4i  8,  16,  32,  64,  &c.  get-metrical  feries  (  eading  ttrms) 
Now  6-f6=i*=the  index  of  the  i  2th   term,  and 
=4°6=the  twelfth  term. 


But,  \vhen  \\\tfrfl  term  of  the  feries  and  the  ratio  are  different,  the 
indices  muft  begin  with  a  cypher,  arid  the  fum  of  the  indices,  made 
choice  of,  mult  be  one  lefs  than  the  number  of  tert/u,  given  in  the 
queftion  ;  becaufe  i  in  the  indices  (land?  over  the  fccond  term,  and  z 
tn  the  indicts,  over  the  third  term,  &c.  And,  in  this  cafe,  the  ftuduti 
of  any  tivo  terms,  divided  by  they///?,  is  equal  to  that  term  t-nyond 
the  fix  ft,  tignified  by  the/^OT  of  their  ind'nes. 


Thus      ^°'  Ij  2>     3'     4|       5'       6)  &C    indices' 

'     Ci>  3>  9>  Z7>  8l^  243.  7^9.  ^c.  geometrical  feries. 

Here,  6  +  5  =          n   tht  index  of  the   I2th  term 

7  29X143  =  I77  J47  {he  lath  ferm,  becaufe  the  firO  term  of 
the  fxrics  auu  trie  ratio  are  different,  by  which  mean  a  cypher  flantls 
ovi  r  the  fir  (I  term. 

THUS,  by  the  help  of  thefe  indices,  and  a  few  of  the  firft  terms 
in  any  geometrical  kries,  any  term,  whofe  diltance  vrom  the  firft 
tt;rm  it.  affigned,  though  it  were  ever  fo  remote,  may  be  obtained 
wuhout  producing  all  the  terms. 

NOTE.  If  the  ntto  of  any  geometrical  feries  be  double,  the  dffir- 
ence  of  the  gteatijl  and  leaf.  ttrmsi.-.-eqiul  to  the  fum  of  ali  the  terms, 
ex  ept  the  Created  ;  if  the  ratio  be  triple,  the  difference  is  double  the 
fum  of  ali  "but  the  iire<neft  ;  if  the  ratio  be  quadn;plet\.\\z  difference 
is  triple  the  fum  of  ail  hut  the  greatcft,  &c 

In  -iny  (cries  in  -H-  ckcreafing  to  infinity  —  if  the  fjuare  of  the 
Jirjl  term  be  divided  by  the  difference  between  the  jirjl  and  fccond^ 
ihe  quotient  will  be  the  fum  of  the  feries, 

R 


194       GEOMETRICAL  PROGRESSION. 

i,     2.     3,       4+     4=         8 

4,    16,  64    2  5  6"x  2  5  6=6  5;  5  3  6= power  of  the  ratio,  whofe 

exponent  :;  leis  by  i.  than  the  number  of  tc-n;?. 

65536=81}?  power  of"  the  ratio. 
Multiply  by         4=fii  ft  term. 

262  i44=la(l  term. 
Or,   4X4 8 =2 62 144=1  he  anfwer. 

2.  If  the  lait  term  be    262144,   the  ratio    4,   and  the 
Lumber  of  terms  9,  what  is  the  firft  term  ? 

Lait  term. 

8th  power  of  the  ratio  48=65536)262 144(4=0X1  term. 
262144 

Or,  =4  the  firft  term. 

4s 

Aga.  Given  the  firft  term,  and  the  ratio  to  find  any 
other  ten  afllgned 

RULE    I. 

When  t%e  indices  begin  *witb  an  unit. 
i.  Write  down   a  few  of  the    leading    terms  of  the 
feries,  and  place  their  indices  over  them. 

2  Add  together  fuch  indices,  wbofe  fum  iliall  make 
tip  the  entire  index  to  the  term  required. 

3.  Multiply  the  terms  of  the   geometrical   feries,  be~ 
longing  to  thofe  indices,  together,  and  the  product  will  be 
the  term  fought. 

EXAMPLES. 

i.  If  the  firft  term  be  2  and  the  ratio  2,  what  is  the 
13th  term  ? 

1,  2..  3,     4,     5+  5+3=     13 

2,  4,  8,   1 6,  32X32X8=8192  Anfwer. 
Or,  2X2l2=8i92. 

2.  A  merchant  wanting  to  purchafe  a  cargo  of  horfes 
for  the  Weft-Indies,  a  jockey  told  him  he  would  take  all 
the  trouble  and  expenie,  upon  himfelf,  ti  collecting  and 
purchafing  30  horfes  for  the  voyage,  if  he  would  give 
him  what  the  laft  horfe  would  come  to  by  doubling  the 
whole  number  by  a  half  penny,  that  is,  two  farthings  for 
the  firft,  four  for  the  fecond,  eight  for  the  third,  &c  to 
which  the  merchant,  thinking  he  had  made  a  very  good 


GEOMETRICAL  PROGRESSION,        i9S 

bargain,  readily  agreed  :  Pray,  what  did  the  lafl  horfe 
come  to,  and  what  did  the  horfes,  one  with  another,  coit 
him  ? 

1,  2,  3,     4,     5,     6+  6=izth          12-f     12-f    6=laft  ter. 

2,  4,  8,    16,  32,  64X^4=4096, & 409  5x409^x64= 
1073741 S24qrs.=£i  118481    is    4d.   and   their  average 
price  was  ^"37282    14=.  otd,  apiece. 

RULE   II. 
When  the  indices  begin  with  a  cypher. 

1.  Write  down  a  few  of  the  leading  terms  of  the  feries, 
as  before,  and  place  their  indices  over  them 

2.  Add  together  the  moft  convenient   indices  to  make 
an  index,  lefs  by  j    than  the  number  expreffing  the    place 
of  the  term  fought. 

3  Multiply  the  terms  of  the  geometrical  ftries,  to- 
gether, belonging  to  thofe  indices,  and  muke  the  product  a 
dividend. 

4,  Raife  the  finl  term  to  a  power,  whofe  index  is  one 
lefs  th.in  the  number  of  terms  multiplied,  and  make  the 
refult  a  divifor,  by  which  divide  the  dividend,  and  the. 
quotient  will  be  that  term  beyond  the  finl,  fignified  by 
the  Aim -of  thofe  indices,  or  the  term  fought. 

3.  If  the  fir  it  term   be   5   and  the.  ratio  3,  what  is  the 
7th  term  ? 

o,  i,  2,  3+  2+  i=  6=tndex  to 6rh  term  beyond 
5,  15,  4J,  »  35X45X1  5  =9 1  i  25=^>ivijend.  [the  ift  or  7th. 
The  number  oi  term^  multiplied  is  3,  (viz.  135X3$* 
15)  and  3 — 1=2  is  the  power  to  which  the  term  5  is  to 
be  rai'cd  ;  but  the  fecond  power  of  5  is  5X5=25,  and 
there-fore  91 125-7-25=3645  the  jihtcnn  required. 

PROBLEM    IT. 

Given   tb;  firjl  term,  the  rath,  nnl  numjjr  of  tcrms,  to  find 
the  f'.im    of  th;  feries. 

RULE.  Raife  the  ratio  to  a  power  whofe  index  fhall 
be  equal  to  thi  number  of  term^,  from  which  fubtraft  i  ; 
divide  the  remainder  by  t)ic  ratio  lefs  i .  and  the  quotient, 
multiplied  by  the  firit  t.^rm,  will  give  the  fum  of  the 

feries. 


i9<5        GEOMETRICAL  PROGRESSION. 

EXAMPLES. 

1.  If  the  Brft  term  be  ?,  the  ratio  3,  and  the  number 
of  terms  7,  what  is  the  ftim  of  the  feries  ? 

Subtract         i         [the  ratio. 
Divide  by  the  ratio  les  1=3— -1=2)2186 

Quotient=ic93 
Multiply  by  the  firit  term=       5 

Sum  of  the  feries  =5465 

Or, X  5=5  465  AnSc 

2.  What  debt  can  be  difcharged  in    a  year,  by  paying 
i  cent  the  6r ft  month,    10  cents  the   fecond,   and  fo  oa, 
each  month  in  a  tenfold  proportion  ? 

IO12  —  I-f-IO— IXJ  — 1  I  I  I  1  I  I  I  I  I  I  1C. = 

D.i  i  ii  1 1  ii  i  T    nc.  Ans. 

3.  A  man  threfhed  wheat  9  days  for  a  farmer,  and  agreed 
to  receive  but  8  wheat  corns  for  the  fir  (I  day's  work,  64  for 
the  fecond,  and  fo  on  in  an  eight  fold  proportion  ;    I  de- 
mand what  his  9  days'  labour  amounted  to,  rating  the 
wheat  at  53.  per  buihel  I* 

89— i 

X  8^=153391688  corns.       Amount=>^78  os.  5$J, 

4.  An  ignorant  fop  wanting  to   purchafe    an  elegant 
houfe,  a  facetious  gentleman  told  him  he  had  one   which 
he  would  fell  him  on  thefe  moderate  terms,  viz.  that   he 
iliould  give  him  a  cent  for  the  firft  door,  2  cents  for  the 
fecond,  4  cents  for  the  third,  arid  fo  on,  doubling  at  every 
door  which  were  36  in  all  ;   it  is  a  bargain,  cried  the  iim- 
pleton,  and  here  is   a  guinea  to  bind  it  :    Pray  what   did 
the  houfe  colt  him  ? 

X  1=68719476735^=0. 687 194767  35c.  Ans. 

2 — i 

*  Note,  7680  wheat  or  barley  corns  are  fuppofed  to  make  a  pint, 


GEOMETRICAL  PROGRESSION.        197 

5.  Suppofe  one  farthing  had  been  put  out  at  6  per 
cent,  per  annum,  compound  intcreil.*  at  the  commence- 
ment of  the  Chriftian  aera  :  what  would  it  have  amounted 
to  in  1784  years  ;  and  fuppofe  the  amount  to  be  in  ftand- 
ard  gold,  allowing  a  cubick  inch  to  be  worth  53!.  2s.  8<UV 
how  large  would  the  mafs  have  been  ? 

2  1  -5  O j 

Ans. X   i  = 

2 — i 

£1486716346»687482094357145515G9S90767065361  11s.  3|d, 
=27980859722121230415979571232933594210766  Cllbick  inches 

of  gold. 

As  355  :    n3   ::  360X69-5   :  7 964 earth's  diameter. 
360x69-5X7964X1327-33  =  264482820122    cubick    miles 

in  the  globe. 
.—6727343730885471136^832000  cub. Inches  in  the  globe. 

Then,  279808597221 21 23041597957 [2329335942 10766 

-^-67  2  7  33  3  7  3088  5474 1 36883  2000 

=4 i 5030899*402 88  8,  which,  however  incredible  it  may 
appear  to  fome,  is  more  than  four  hundred  and  fifteen 
millions  of  millions,  nine  hundred  and  thiity  thoufnnd 
eight  hundred  and  ninety  nine  millions,  eight  hundred 
and  forty  thoufand  two  hundred  and  eighty  eight  times 
larger  than  the  globe  we  inhabit. 

PROBLEM    III. 

The  firft  term,  the  Jaft  ferm,   (or  th?  txtrsmst)  and  the  ratio 
given,  to  fnd  the  fum  of  the  ferhs. 

RULE.  Divide  the  difference  of  the  extremes  by  the 
ratio  lefs  by  i  :  Add  the  greater  extreme  to  the  quotient, 
and  the  refult  will  be  the  fum  of  all  the  term*. 

Or,  Multiply  the  greateft,  term  by  the  ratio,  from  the 
produft  fubtiacl  the  ieaft  term  ;  then  divide  the  itma^n- 
der  by  the  ratio,  lefs  by  J5  and  the  quotient  will  be  the. 
fum  of  all  the  terms. 

f  Any  fum,  at  61.  per  cent,  per  annim,  compound  intercft,  \vill 
dou'ilcin  1 1  years  and  three  hundred  and  twenty  live  days,  or  11-889. 
y.  4;  s.  or  1 1  8y  is  ne-ir  enough  ;  th^n  if  you  divide  1784  by  <  I  89  it 
•will  -;ivc  the  number  of  terms  in  ihii  caie  cqiial  to  J^o  :«— The  r«aio 
\vili  be  2,  and  the  firft  term  x. 
R  2 


i98       GEOMETRICAL  PROGRESSION. 

Or,  When  all  the  terms  are  given,  then,  from  the  pro- 
duct of  the  fecond  and  laft  terms  fubtracT:  the  fqiure  of 
the  firft  term  ;  this  remainder  being  divided  by  the  fecond 
term  lefs  the  firft,  will  give  the  fum  of  the  feries. 

EXAMPLES. 

i.  If  the  feries  be  2,  6,  18,  54,  162,  486,  1458,  4374, 
what  is  its  fum  total  ? 

Firft  Method. 

From  the  greateft  term=4374 
Subtract  the  leaft  =       2 

Divide  by  the  ratio  —  1=3—1=2)4372  difF.  of  extremes* 

Quotient^  2   86 
Add  the  greater  extreme-43-4 

6560 


Second  Method. 

Greateft 
Multiply  by  the  ratio= 

Produ  01=131  22 
Subtract  the  leaft  term=         2 

Btvide  by  the  ratio  lefs  by  1=3  —  1=2)13120 


4374><3~2 
Or,  ---  =6560. 

3~' 

Third    Method. 

Greateft 
Multiply  by  the  fecond  term=       6 


Subtract  the  fquare  of  the  firft  term=2X2= 

Divide  the  remainder  by  the    fecond    1      r  %-  ^ 

term  lefs  the  Hrft,  }  =6-2=4)2^240 


Ans. 


GEOMETRICAL  PROGRESSION        199 


4374x6—4 

Or, =6560. 

6—2 

2.  A  m.m  travelled  6  days  ;   the  fifft  day  he  went  4 
milev,  and  each  day  doubling  his  day's  travel,  his  latl  day's 
ride  was  128  miles  :   How  far  did  he  go  in  the  whole  ? 

128—4 

\-  i  28=252  miles. 

2 — i 

3.  A  gentleman  dying,  left  5  fons ;   to  whom  he- be* 
queathed  his  eltate  as  follows,    viz,    to  his   youn^e  t    fon 
loocl.  to  the  eldeft  5062!.  EOS.  and  ordered  that  each  fon 
fhould  exceed  the  next  younger  by  the  equal  ratio  of  r-i  : 
What  did  the  feveral  legacies   amount  to  ? 

5c62'5 — -tcoo 

— h  5062 -j^"  131 87  Jos.   Ans> 


PROBLEM     IV. 
the  extremes  and  ratio,  to  find  the  number  of  ttrmi. 

RULE. 

Divide  the  greateft  term  by  the  leaft  ;  find  what  power 
cf  the  ratio  is  equ-ii  to  the  quotient  ;  then,  add  r  to 
the  index  of  that  power,  and  the  fum  will  be  the  number 
of  terms 

Or,  Subtract  the  logarithm*  of  the  leaft  term  from 
th  u  of  the  greateft  ;  divide  the  remainder  by  the  loga- 
rithm of  the  ratio,  and  add  i  to  the  quotient. 

EXAMPLES. 

i.  If  the  leaft  term  be  2,  the  greateft  term  4374>  and 
the   ratio  3,  what  is  the  number  of  terms  ? 
Divide  by  the  leaft  ternj=i)4374=g.reatell  term. 

3x3x3x3x3x3x3=qaotisnt»  2i8y=7ih  power   of  the  ra- 
tio ;  then  7+1=8    -ins. 

*  Logan th--  arc  artliir;al  numbers  the  addition  of  which  anfvrer.< 
:o  multiplication  of  whole  number?,  and  I'ubtrd&icn  to  divifiou. 


700       GEOMETRICAL  PROGRESSION. 

Or   from  the  logarithm   of 

the  greats  ft  term  =  3  64088 

Subtree!  the  logarithm   of 

the  Seatt  term  e=  0-30103 

Divide  the  rerrm.  1  » 

by  the  logarithm  >  £=.-47712)3  33985(7+1=8,  Anfwer. 
of  the  ratio  j  3  3*984 

i 

2.  A  gentleman  travelled  252  miies  ;  the  firft  day  he 
t'ode  4  miles,  the  laft  128,  and  each  diy  s  j -urn^  was 
double  to  the  preceding  one  :  How  many  days  was  he  in 
performing  the  journey  ?  Ans.  6  days. 

PROBLEM    V. 
Givsn    the  leaft  term,  the  ratio    and  the  fum   of  the  feries,  to 

find  the  Lift   term. 

RULE.  Multiply  the  fum  of  the  feries  by  the  ratio,  lefs 
l  ;  to  that  product  add  the  fitil  teim,  and  the  refult,  divi- 
ded by  the  ratio,  will  give  the  la'.t  term. 

EXAMPLES. 

If  the  i  ft  term  be  2,  the  ntio  3,  and  the  fum  of  the 
feries  6560  ;  what  is  the  Uft  term  ,? 

Sum  of  the  feries=656o 
Multiply  by  the  ratio  lefs  i  =       2 

Produ<5t=i3  *  20 
Add  the  leaft  term=         2 

Divide  their  fum  by  the  ratio=3)  '3122 

4374  Axis, 
3 — 1x6560-1-2 

Or, =4374  Ans. 

3 

2.  A  gentleman  performed  a  journey  of  252  miles; 
the  firiT  d.iy  he  rode  4  miles,  an  I  each  dav  ittcr  the  firft, 
twice  fo  fir  as  the  d^y  before  :  How  far  did  he  ride  the 
laft  day? 

2 iX*52.+4 

=  iz8  miles,  Anfwer, 


SIMPLE   INTEREST.  301 

SIMPLE  INTEREST. 

INTEREST  is  a  premium  allowed  by  t lit!  borrower 
to  the  lender,  according  to  a  certain  rate  per  cent,  agreed 
on  ;  which  by  law  is  ftated  at  6  per  cent  per  annum. — » 
Principal  i&  the  money  lent.  Rate  is  the  furn  per  cent,  a- 
greed  on  Amount  is  the  rum  of  principal  and  intereft. 

Simple  Intereft  is  that,  which  is  allowed  on  the  princi- 
pal only. 

NOTE.  By  this  rule,  Commiffion,  Brokerage,  Infur- 
ance,  Furchafmg  Stocks,  or  any  thing  eiie,  rated  at  fo 
much  per  cent,  are  calculated. 

GENERAL   RULE. 

i.  Multiply  the  principal  by  the  rate,  arid  divide  by 
100,  (or  cut  off  the  two  right  hand  figures  in  the  pounds) 
and  the  quotient,  or  left  hand  figures,  will  be  the  anfwer 
in  pounds,  &q  the  right  hand  figures  being  reduced  and 
cut  off  as  at  firft.  If  the  principal  bs  dollars,  the  right 
hand  figures  will  be  cents. 

2  For  more  years  than  one,  multiply  the  istereft  of 
one  year  by  the  number  of  years. 

3.  For  any  number  of  months,  t«".Ve  the  aliquot  parts 
of  a  year ;  and  for  days  the  aliquot  parts  of  30. 

f9l    v  >,  fll 

NOTE.  When  [  8  |  -5  jj  |  f  of  the  given  number  of 
the  rate  per  cent,  j  6  I  ^"§_  \  £  \  months,  and  you  will 
per  annum  is  ]  4  j  •§*'«  j  j  \  have  the  intereft  for  the 

I  3  i  I  -|  I  i  I  give»  timer 

EXAMPLES. 

i.  What  is  the  intereft  of  573!.  135.  9^-  at  6  per  cent, 
per  anr.um  ?  An.s.  ^34  8s.  jd, 

£573     '3     9£ 


202 


SIMPLE  INTEREST. 


2.  What  is   the   intereft   of  ^329    175.    6jd.    for    3 
years,  7  months,  and  12  days,  at  5  per  cent,  per  annum  ? 

17     6i 


i-6|49       7     81 

20 

"9187 

12 

1C|$2- 

4 

2 

10 

Then, 
6  mo.     £     i< 

5    9     ic%  intereft  of  i  year. 
3 

49     9       7?  do.  of  *  years, 
i  mo.     j-       8     4     i  r^  do  of      6  months, 
10  da.      -j       *    "7       5  1  4°-  °f       '  month. 
2  da.       ^             9       i|  do.  of           10  days. 
i       9!  do-  of             2  days. 

Ans.  £59  13             do.  of  ^y.  701.  lid. 

Or  thus  : 
6  months 

£     s.    d. 

A     329     '7     6-J- 

i        16       9   loj 
3 

j  i   month 
10  days 
i  days 

1 

49       9     7i 
1         8       4   lit 

T         l       7     5i 
i                 9     'i 
i     9i 

SIMPLE  INTEREST.  203 

3.  What  is  the  intereft  of  347  dollars   50  cents,  at  6 
per  cent,  per  annum  lor  a  year  ? 

1^347  5° 
6 


2o  8500  Ans.  Dao   85c. 

4.  What  is  the  intereft  of  0797    i3c.  at  6   per  cent. 
per  annum,  for  8  months  ? 

1)797-13 

4 



31-8*52  Ans.  D3i   88c.  5T\m. 

5.  What  is  the   intereft  of  1)649  1 7c.  at  6  per  cent, 
per  annum,  for  15  months  ? 

17 
7i 


454419 

3*45*5 

48  68775  Ans.I)48  68c.7^m, 

6.  Required  the  amount   of  ^"725    125,.    6d.  at  5  per 
cent,  per  annum,  for  a  year. 

5=^)7*5     12     6 

36       5     7* 


Ans.  £761      1  8     if 

7.  What   is  the  amount  of  Dj6o  5oc.  at  6  per  cent, 
for  16  months  ? 

0560    50 
8 

44-84 
56^-50 


Ans. 

8.  What  is  the  intereft  ot  Dijo  7j;c.  for  I  month,  at 
6  per  cent,  per  annum  ? 


204  SIMPLE  INTEREST, 

1)150-75 

•75375     Ans.  75c. 
So  that  any  number  of  dollars,  confidereci  as  fo  many 
cents,  is  the  interelt  for  2  months,  at  6  ptr  cent. 

COMMISSION,    OR  FACTORAGE, 

Is  an  allowance  of  fo  much  per  cent,  to  a  Factor  or 
Correfpondent,  for  buying  and  felling  goods. 

9.  Required  the   commiffion  on   £436  95.  6d.  at  3^ 
per  cent. 

£436     9     6 
31 


i  309     8     6 
218     4     9 


20 
5153 

12 


JJ56  Ans.  £15  5  Q. 
io»  Required  the  commiffion  on  D649  yjc.  at  if  per 

«at-\t-  '        I      f\AC\'1P 


cent.  I  I  649-75 

I  3H-875 
162-4375 

1137-0625  Ans.  DII 


BROKERAGE, 

Is  an  allowance  of  fo  much  per  cent,  to  a  perfori  caU 
Jed   a  Broker,  for   afliiting  merchants  in  purchafui^  or 
goods. 


SIMPLE  INTEREST.  205 

n.  Required  the  brokerage  on  ,€911    T2s.  at  53.  or  -^ 
per  cent. 

12 


2\27       IS 
20 


12 

.  —  —         Ans.  £2  55;  6jd. 
6|96 
4 


12.   Required    the  brokerage   on   0876  2ic.   at  33^ 
cents,  or  at  -s  per  cent 


292-07     Ans.  D2  920. 


BUT  ING  AND  SELLING  STOCKS. 

Stock   is  a   general  name  for  the  capitals  of  trading 
Companies. 

13.   Required  the  amount  of  £375   155.  bank  flock,  at 
£75  percent. 


5° 


375      15  Or  thus  : 

1 I  25  I  i  I  375      '5 

18       i?  6  — * 


93      18  9  Subtract     93      18     9 

Ans.  £281      1 6  3  As  before,  £281      16     3 

14.  Required  the  amount  of  D2I95  500.  bank  ilock, 
at  125  per  cer.t. 

I  ^5  I  4  I  2i95'5<> 
Add          548-875 

Ans.  Dz744  375 


SIMPLE  INTEREST 


SIMPLE  INTEREST  BT  DECIMALS. 
A  TABLE  of  Ratios,  from  one  pound  to  ten  pounds* 


Rate  per 

cent. 

Ratios. 

Rate 
pr.ct 

Ratios. 

Rate 
pr.ct. 

Ratios. 

I 

•01 

3 

•03 

5 

•05 

,  JL 

4 

•0125 

3t 

•0325 

5* 

•©55 

ly 

•015 

3* 

•C35 

6 

•c6 

T    3 

'4 

•0175 

kl 

•0375 

6^ 

•065 

2 

'02 

4 

•04 

7            "07 

** 

•0225 

4i 

•0425 

8            -c8 

4- 

•02J 

4* 

•045 

9 

•09 

*! 

•0275 

4f 

•047.? 

10 

•i 

Ratio   is  the  Simple   Intereft  of  £  l  or  Dollar,  for,   I 
year,  aC  the  rate  per  cent,  agreed  on. 

A  TABLE  for  the  ready  finding  of  the  decimal  parts 
of  a  year,  equal  to  any  number  of  dayt,  or  quarters  of 
a  year. 


Days 

dec.  pts.  j  Days. 

dec.  pts.  j  Days 

dec.  pts. 

i 

2 

3 
4 

56 

7 
8 

i  9 

•00274 
•005479 
0082  19 
•0,0959 
013699 
•016438 
•019178 
•021918 
•024657 

10 

20 

3P 

40 

50 
60 
70 
80 
90 

•027397 
•054794 
•082192 
•109589 
•136986 
•164383 
•191781 
•219  [78 
•246575 

100 
200 
300 
400 

•*7*973 
•547945 
•821918 

I  'OOOCOC 

\  of  a  year=-25 
J-  of  a  year='5 
^of  a  year=  75 

CASE    I. 

The  principal)  time,  and  ratio  givffn9  to  find  the  inlerejl  and 
amount. 

RULE.  Multiply  the  principal,  time,  and  ratio  con- 
tinually together,  and  the  laft  prcdud  will  be  the  inter- 
eft,  coxnmiflion,  brokerage,  &c  to  which  add  die  princi- 
pal, and  the  fum  \\ill  be  the  amount. 


BY  DECIMALS. 

EXAMPLES. 

i.  Required   the   amount  of   £^37    XOSt  at  £&   Per 
Cent,  per  annum,  for  5   years  ? 

Principal  537*5 
Multiply  by  the  ratio=  «  »o6 

Produft  32-250 
Multiply  by  the  time=          5 

Intereft=i6i  "250 
Add  the  principal  =5  3  7  -5 


20 

15-00  Ans.  £698  15$. 
®r>  537  5X06x5  -f537'5=r;£698   155. 

2.  What  is  the  fimple  intereft  of  £917   i6s.  at  £5  per 
cent,  per  annum,  for  33-  years  ?  Ans.  £321   4$.  jd. 

3.  What  is  the  amount  of  £235   35.  gd.  at  £5%  per 
cent,  per  annum,  from  March  5th,  1784,  to   November 
23d,   1784?  Ans.  £244  o  8i. 

4.  Tf  my  correfpondent  is  to  have  ^"2!  per  cent,  what 
will  his  commifuon  on  .£785    155.  amount  to  ? 

Ans.  £19   12    i  civ 

5.  If  a  broker   difpofes  of  a  cargo  for  me,  to  the  a. 
mount  of  £637   IDS.   on  commifllon   at   £i;£  per  cent. 
and  procures   me    another  cargo  of  the  value  of  /"8  17 
1  55.  OR  commiiHon  at  £\^  per  cent.  ;  what  will  his  com- 
miffion  on  both  cargoes  amount  to.  ?         Ans   £22  5  7. 

CASE  II. 

The  amount,  tin:?t  and  ratio  gi<usny  to  find  the  principal. 

RULE. 
Multiply  the  ratio  by  the  time  ;  add  unity  to  the  pro- 


,  for  a  divifor,  by  which  fum  divide  the  amount,  and 
the  quotient  will  be  the  principal, 


SIMPLE  INTEREST 

EXAMPLES. 

i  What  principal  will  amount  to  ^1045  !4St  'm  7 
years,  at  £6  per  cent,  per  annum  ? 

Ratior?  06 
Multiply  by  the  time=t    y 

Product  =2*4* 
Add    !• 

Divifor=  i '42)1045-7(736-4084^  =£7  36  8s* 
sd.  anfwer. 

1045-7 

Or,  =£736  8s.  2d.  Ans. 

•06x7+1 

2.  What  principal  will  amount  to  £3810  in  6  years, 
at  £4$  per  cent,  per  annum  ?  Ans.  £3000. 

3.  What  principal  will  amount  to  £666  95.  c£d.  in  3-^ 
years,  at  £5-^  per  cent,  per  annum  ?  Ans,  £563, 

CASE   III. 

7 he  amount)  principal,  and  time  given,  to  find  the  ratio. 
RULE. 

Subtract  the  principal  from  the  amount ;  divide  the 
remainder  by  the  product  of  the  time  and  principal,  and 
the  quotient  will  be  the  ratio. 

EXAMPLES. 

1.  At  what  rate  per  cent,  will  £543  amount  to  £705 
•?#s.  in  5  years  ? 

From  the  am ount=7 05:^9 
Take  the  pr:ncipal=r543 

Divide  by  543X5=27i5)i62'9o(  06 

162-90 
-05-9—543 

Or, =-06=^6  Anfwer. 

543^5 

2.  At  what  rate  per  cent,  will  £391    175.  amount  to 
M49  3S-  i|d.  '74qr.  in  sJ  years  ?  Ans.  £4$, 

3.  At  what  rate  per  cent,  will  £413  125.  6d  amount 

45.  io£d.  in  4|  years  ?  Ans.  ^*6|. 


BY  DECIMALS, 

CASE  IV. 

t*The  amount,  principal^  and  rate  per  cent,  given^  to  find  the  time, 

RULE. 

Subtract  the  principal  from  the  amount  ;  divide  the 
remainder  by  the  product  of  the  ratio  and  principal,  and 
the  quotient  will  be  t-he  time. 

EXAMPLES. 

1.  In  what  time  will  £543  amount  to  £705    i8s.  at 
£6  per  cent,  per  annum  ? 

From  the  amountrryoj^ 
Take  the  principals:  543 

Divide  by  543X'o6=32 -58)  162-9(5  years,  An?. 

162  9 

2.  In  what  time  will  ^"3000  amount  to  .£3810,  at  £+£. 
per-  cent,  per  annum  ?  Ans.  6  years. 

3.  In  what  time  will  £391    175.  amount  10^441    33. 
i|-d.  at  £4!  per  cent,  per  annum  ?  Ans.  3^-  years. 


To  fmd  ths  inter  eft  of  any  fum  at  6  per  cent,  per  annum,  for 
any  number  of  months : 

RULE. 

If  the  months  be  an  even  number,  multiply  the  prin- 
cipal by  half  that  number  ;  and  if  the  months  be  un- 
even- halve  the  even  months,  to  which  annex  T5^  ;  thus 
the  half  of  19  is  9*5  ;  and  multiply  the  principal  as  be- 
fore, cutting  off  two  figures  more  at  the  right  hand, 
than  there  are  decimals  in  both  factors,  which  reduce  to 
farthings,  each  time  cutting  off  as  at  firft. 

4.  What  is  the  intereft  of  ^"349    »6s.  6d.  for  9    years 
and  1 1  months,  at  6  per  cent,  per  annum  ? 
Y.  m. 
9     ^ 

12 


S  z 


2)1 19  months. 

59-5=2  number  of  months. 


SIMPLE  INTEREST 


Principal  345*825 
Mult,  by  -J  number  of  Mo.         59-5 

1729125 

3«H*5 

1729125 


^205-765875=^205   15  3, 
'          16  6 


u 


A  TABLE  of  decimal  parts  for  every  day  in  the  twelfth 
part  of  a  year,  which  confifts  of  365^  days. 


D. 

dec. 

D. 

dec. 

D 

|  dec. 

D. 

dec. 

D. 

dec. 

;    I 
2 

3 
4 

5 

6 

033 
•066 
•098 
•131 
•164 
•197 

7 
8 

9 

10 

1  1 

12 

•230 
263 
•296 

•3*8 

•36, 

'394 

M 
if  4 

15 
16 

17 

i8 

•427 
•460 

•493 
•526 

•558 

•591 

'9 

20 

21 
22 

23 

24 

•624 
•657 
•690 

•723 
•756 
•788 

25 

26 

27 
28 
29 

3.0 

•821 
•854 
•887 
•920 

'953 
•986 

NOTE.     This  Table  may  aJfo  be  ufed  for   ihe  parts  of 
a  year>    in   Compound  Intereft:  after   having  worked  for 

whole  years. 

^Te  Jitid  ins    Inisreft-  of   any    juni,  either  for    months^    er 
months  and  dajsy  at  6  per  csnl.  per  annum* 

RULE. 

Multiply  the  principal  by  the  number  of  month?,  (or 
months  and  parts,  anfwering  to  the  given  number  of  days 
in  the  table)  and  cut  off  one  figure  at  the  right  hand  of 
the  produft  more  than  is  required  by  the  rule  in  decim- 
als, and  the  product  will  be  the  intereft  for  the  given 
lime,  in  {hillings  and  decimal  parts  of  a  (hilling. 

EXAMPLES. 
i.  What  is  the  intereft  of  j£ico  for  a  year  ? 

Principal=ioo 
Multiply  by  the  months=   1 2 


Ans,  Si 


IN  FEDERAL  MONEY.  2ri 

2.  What  is  the  intereft  of  £250    los.  for   19  months 
and  7  days  ? 


Ans.  5481-71  15=^*24  i   8^ 

Another  method  of  calculating  IniercJ}  for  Months,  at  £6  per 

cent,  per  annum. 

RULE. 

If  the  principal  confift  of  pounds  only,  cut  off  the  unit 
figure,  and,  as  it  then  ftands,  it  will  be  the  intereft  for  i 
mo-nth  in  (hillings  and  decimal  parts  :  —  If  it  confift  of 
pounds,  (hillings,  &c.  reduce  the  (hillings,  £c.  to  decim- 
als, which,  with  the  unit  figure  of  the  pounds,  will  be 
decimal  parts  of  a  (hilling. 

EXAMPLES. 

i.  What  is  the  intereft  of  2.  What  is  the  intereft 
£\"1$  for  5  months  ?  of  255!.  i6s.  for  7  months  ? 

£  175=  17  '5J».=:  intereft  for  i  (hi!.       [month. 

month.  .     175    j£*55   16=25*58  int.  for    i 

Multiply  by  the  tirnezz      5  7 


Ans   £476  /*8   19  c£  Ans. 

SIMPLE  INTEREST  IN  FEDERAL.  MONET. 

PROBLEM   I. 

When  ffc  principal  is  given  in  Mxffachufitis  pounds,  Jh  tilings, 
£sV.  and  the  inter^Jl  ;'/  retired  in  Federal'  MQUQ,  at  6  psr 
cent,  per  annum. 

RULE. 

Reduce  the  (hillings,  £c  to  their  equivalent  decimal, 
divide  the  whole  by  5,  and  the  quotient  is  the  annual  in- 
tereft :  Or,  multiply  the  principal  by  2,  and  the  pr 


SIMPLE   INTEREST 

(having  the  unit  figure  of  the  pounds  cut  off,)  will  be 
the  intereft  as  before. 

EXAMPLES, 

i.   Required   the  annual  intereft  of  5  1  7!.  35.  7|d.  at 
6  per  cent. 


7id.=-c>3o-  --  D.    c.    m. 

Excefs  of  12    =-coi  io3»4i6=ic3  43  6    Ans* 

•i  8-1         Or,  517-181 
2 


103-4362=0103  430.  6TViu.. 

2*  Required  the  annual  intereft  of   il.  in  cents. 
5)1-00 

20  cents,  Aus. 


PROBLEM   II. 

fflben.  the  principal  is  given  in  Majjfachufstts  old  currency^ 
and  the  Interejl  and  amount  are  required  in  Federal  Money 
at  6  per  ccni. 

RULE. 

Reduce  the  Mafiachufetts  money  to  Federal,  then  di- 
vide the  principal  by  20,  and  that  quotient  by  5  ;  add 
thofe  quotients  together,  and  they  are  the  intereft  ;  or 
add  them  to  the  principal,  and  they  are  the  amount. 

EXAMPLES. 

i.  Required  the  amount  of  425!.  i6s.  S^d.  for  i  year? 
at  6  per  cent. 


•034        20)1419-450 
•ooi  5)70-9725 


i6s.  8d.=^'  835  ---    D,      c.  m. 

15046170=1504  61   7  Ans* 

2.  Required  the  amount  of  £112,  45,  6d,  for  i  ysaft 


IN  FfcDER^L  MONEY.  213 

•2  3)11*225 

•034  20)3-4083 

•ooi  5)18*7^41 

-  ,  3-7408 

45.  6d.=r^*'225  —  ---     D.  c.  m  dec. 

39652  79~396'52  7    9 


PROBLEM    III. 

When  the  principal  ir    Majfacnnfetts  old  currency^   and  the 
monthly  interefl  is  required  in  Federal  Money. 

RULE. 

Reduce  the  (hillings,  &c.  to  decimals,  then  feparate  the 
right  hand  figure  of  the  pounds  with  the  decimals,  divide 
by  6,  and  the  quotient  is  the  anfwer  in  dollars,  cents,  &c. 

EXAMPLE. 

Required  the  monthly  interefl  of  425!.  i6s.  8£dv  ip 
Federal  money  ? 

•8  6)42-5835 

•034  --  D.  c.  m. 

•ooi,  7  -09  7  25  =  7  -09  7£  Ans. 


PROBLEM     IV. 

When  the  principal  is  Federal  Monty,  and  the  inter  ejl  is  re^ 
guired  in   the  fawe. 

RULE. 

Work  according  to  the  general  rule  in  fimple  intereft, 
that  is,  multiply  by  the  rate  of  interefl,  feparate  the  two 
right  hand  Hgures  of  the  dollars  in  the  prqducl,  and  it 
will  give  the  intereft  in  dollars,  cents.  &c. 

N  B.  The  figures,  which  are  more  than  three  places 
to  the  right  hand  of  the  point,  are  of  no  account,  unlefs 
the  fourth  place  exceed  5,  in  which  cafe  increafe  the 
mills  f. 

EXAMPLES. 

T.  What  is  the  annual  intereft  of  0537  240*.  6m.  aX 
6  per  cent.  ? 

537-24  6 
6 

—  --    D.  c.  m. 
32-23476=32-23  5  Ans< 


2  1  4  SIMPLE  INTEREST 

2.  What  is  the  intereft   of  01465    460.  6m.  for    16 
months,  at  6  per  cent,  per  annum  ? 

1  465  '46  6 

8=half  the  number  of  months- 
----     D  c,  m, 
1  17-23728=1  17-2^3  7  Ans. 

3.  What  is    the   interefl  of  0537  340.  7m.  for   19 
months,  at  6  per  cent,  per  annum  ? 

9f5=half  the  number  of  months. 

268673.5 
4836123 

^  ---  D.    c.  m. 
5-1-047965  =  51*04  8  Ans. 

N.  B.  Becaufe  there  are  4  decimals  in  the  multipli- 
cand and  multiplier,  I  cut  off  4  figures  for  them,  and 
two  more  according  to  the  rule. 

PROBLEM    V. 

When  the  principal  is  Federal  Money,  and  the  monthly  intsreft 
is  required  in  ths  fa?ne,  at  6  per  cent,  per  anuum. 

RULE. 

Separate  the  two  right  hand  figures  of  the  dollars,  and 
you  then  have  the  intereft  for  two  mouths  ;  half  of  which 
is  the  monthly  imereft  in  dollars,  cents,  &c.  If  there  be 
but  one  place,  or  figure  of  dollars,  a  cypher  muft  be 
prefixed  to  the  left  hand. 

EXAMPLES. 

i.  What  is  the  monthly  intereil  of  Dp  590.  7m.  at  6 
percent,  per  annum  ?  2)'°9597 

Ans.  -047985—  4c.  8m.  nearly. 

?.  What  is  the  monthly  intereft  of  Dioo  5oc.  5m.  ? 
2)1-00505 

•50252=50^  2jm.  Ans. 


RULES  for  calculating   Interefl  for   Days. 

RULE  I. 

Multiply  the  principal  by  the  given  number  of  days, 
and  that  produft  by  the  rate  on  the  pound  ;  divide  the 
laft  product  by  365  (the  number  of  days  in  a  year)  and 
h  will  give  the  intereft. 


BY  DECIMALS.  2(5 

EXAMPLE. 

What  is  the  intereft  of  360!.    IDS.    for   175  days,  at  6 
per  cent.  : 

'5Xi75X-o6 

—  —  —  =£  io<3'=J£'io  75.  4^d.  Anfwer. 


365 

RULE     II. 

Multiply  the  given  principal  by  the  given  number  of 
Jays,  and  divide  the  product  by  6083,  for  6  per  cent.  ; 
(the  number  of  days  in  which  any  lum  will  double  at  that 
rate)  the  quotient  will  give  the  anfwer. 

EXAMPLE. 

What  is  the  intereft  of  317!.  ios.  at  6  per  cent,  per  an- 
num, for  2  1  o  days  ? 

327-5x210 

-  ---  =£1  i'3c6=£u  6  if  Ans. 
6083 

RULE  for  making  a  divifor  for  any  rate  per  cent. 

Multiply  365  by  ico,  and  divide  the  product  by  the 
rate. 

365x»co 
Thus  for  6  per  cent.  --  =6083  divifcr. 

6 

365X100 
For  5  percent.  --  =7300  divifor,  &c. 

5 

Perhaps  the  mofl  convenient  way  to  calculate  at  6  per 
cent  is  firft  to  do  it  for  5,  and  then  add  one  fifth  of  the 
quotient  to  itfelf  ;  becaufe,  by  cutting  off  the  two  cy- 
phers in  the  divifor,  you  have  to  divide  only  by  73. 

Hence,  when  intereft  is  to  be  calculated  on  cafh  ac- 
counts, accounts  current,  or  any  other  accounts  where  par- 
tial payments  are  made,  or  partial  debts  contracted  ; 
multiply  the  feverai  balances  into  the  days  they  are  at 
interert  and  the  fum  of  thefe  products,  divided  as  above, 
will  give  the  intereft  at  5!.  or  61.  per  cent,  and  for  any 


216 


SIMPLE  INTEREST 


other  rate,  make  the  proper  addition  or  deduction ;    or 
find  a  divifor,  as  before  direfted. 

EXAMPLES. 

i.  On  the  i  ft  of  January  I  lent  450!.  xos.  6d.  which 
I  received  back  in  the  following  partial  payments, 
viz.  on  the  >4th  of  Jan.  57!  i  is.  9d.  ;  on  the  7th  of 
February,  39!  35.  lod  ;  on  the  i9th  of  March.  63!.  ;s. 
3d.  5  on  the  4th  of  April,  45!.  ;  on  the  26th  of  April, 
19!  I2S.  6d.  ;  on  the  I2th  or'  M-iy,  icol.  ;  on  the  loth 
of  Jane,  6ol.  75.  $d.  ;  and  on  the  ift  of  Atiguft,  65], 
IDS  :  What  intereit  is  due  at  6  per  cent,  ? 


Dates.                                         £    s.    d  Da.     Frodufts. 

Jan.     i 

Lent  OR  demand 

450 

1C 

6 

'3 

5856  16     6 

14 

Received  in  part 

55 

I  1 

9 

Balance 

392 

18 

9 

24 

9430  10    o 

Feb.    7 

Received  in  part 

39 

r. 

10 

Balance 

353 

i14 

1  1 

40 

14149  16     8 

Mar.  1  9 

Received  in  part 

63 

i 

2 

Balance 

29 

(y 

9 

16 

4647   16    o 

April  4 

Received  in  part 

45 

c 

o 

Balance 

H5 

9 

9 

22 

5400  14    6 

26 

Received  in  part 

19  12 

6 

Balance 

225  17 

3 

16 

3613   i  6    o 

May  1  2 

Received  in  part 

ICO     C 

c 

Balance 

1/5  17 

3 

29 

3650    o    3 

June  i  o 

Received  in  part 

60 

". 

3 

Balance 

65 

10 

c 

52 

3406    o    o 

Aug.  i 

Received  in  full  of 

....   - 

the  principal 

65 

10 

c 

50155     9  ii 

BY  DECIMALS. 


217 


9  11(6     17     4^  intereft  at  j  per  cent. 
438  i       7     Si 


£8       4  ici  intereft  at  6  per  cent. 


5776 


2.  I  have  given  Peter  Trufty  a  cafh  credit  for  loool. 
in  confequence  of  which,  on  the  izth  of  May,  I  paid 
his  bill  for  250!.  ;  May  27th,  paid  his  draught  for  280!.  5 
June  ift,  he  gave  me  a  bill  on  the  Maflachufetts  bank  at 
light,  for  290!.  ;  July  I7th,  he  paid  me  per  receipt, 
70!.  ;  Auguft  2Otli,  he  drew  for  750!.  at  light ;  Auguft 
31,  he  paid  me  per  receipt,  500!.  ;  Sept.  1510,  he  drew 
at  fight  for  135!.  ;  and  3d  of  October  for  175!.  ;  Oct. 
29th,  he  paid  me,  per  receipt,  250!.  ;  and  Nov.  3d, 
125!.  ;  November  I2th,  he  drew  at  fight  for  375!.  ;  and 
November  i8th,  for  125!.  ;  January  jft,  he  paid  me,  per 
receipt,  290!.;  and  January  2Oth,  2lol.  On  the  i  ft 
ot  March  following,  he  demands  a  fettlement  :  What  is 
then  due  to  me,  intereft  at  6  per  cent.  ? 
T 


SIMPLE  INTEREST. 


Dates. 

f  Days.  Prod. 

May 

12 

Paid  his  bill 

250 

15 

375° 

May 

27 

Paid  his 

draught 

280 

Balance 

530 

5 

2650 

June 

I 

Received 

in  part 

29C 

Balance 

240 

46 

11040 

July 

17 

Received 

in  part 

70 

Balance 

170 

34 

5780 

Auguft 

20 

Paid 

750 

Balance 

920 

1  1 

10(20 

Auguft 

31 

Received 

in  part 

500 

Balance 

420 

'5 

6300 

Sept. 

'5 

Paid 

'35 

Balance 

555 

r8 

9990 

October 

3 

Paid 

175 

Balance 

73C 

26 

18980 

Odober 

29 

Received 

in  part 

250 

Balance 

48c 

5 

2400 

November  3 

Received 

in  part 

125 

Balance 

355 

9,      3J95 

Nov. 

12 

Paid 

375 

Balance 

73° 

jr 

4380 

Nov. 

18 

Paid 

125 

Balance 

855 

44 

37620 

January 

I 

Received 

in  part 

290 

Balance 

56519 

10735 

January 

20 

Received 

in  part 

210; 

March 

I 

Balance 

355 

40    14200 

1  4  [  1  40 

SIMPLE  INTEREST.  a  19 


5) 

Then,  73100)141  i4o(    19     6     8  intereft  at  5  per  cent. 

3    '7     4 


4     o  intereft  at  6  per  cent. 
355     o     o 


£378  4  o  balance  in  my  favour. 
When  cafli  credit*  are  given,  a  balance  fhouldbe  made 
upon  every  tranfadrion,  which  (hould  be  multiplied  into 
the  days  the  firft  leifure  minute  ;  then,  when  the  time  of 
fettlement  comes,  you  will  only  hiive  to  add  up  the  prod- 
ucts, and  divide  as  above,  and  the  account  will  be  finiilied. 

3.   A  owes  B  the  following  fums.  with    the   intereft  on 
them,  at  6  per  cent-  per  annum,  as  follows,  viz,   D6o   for 
7  months,   D 150  for  15  months,  D75  500.  for  9  month?, 
Di45   75?.    for    27    months,   and   1)397     6oc.   for    45! 
months :  \Vhat  is  the  amount  of  principal  and  intereft  I 
D  c.     Mo. 
60      x  7     =     420 
150     Xi5     =  2250 
75  5  x  9     =     679-5 
145-75X27     =  3935  25 
397-60x45  5  =18090-8 


828-85      200)25375.55(126-877  intereft. 
828-85     principal. 

1)955 '727  amount,   Ans. 

NOTE.  I  divide  by  200,  the  number  of  months,  in 
which  any  furn  will  double  at  6  per  cent,  p^r  annum,  and 
it  gives  the  interell. 

When  partial  payments  are  made  upon  bonds,  nore?, 
&c.  al  any  interval  greater  than  a  year,  the  intereft  is  cal- 
culated in  a  progreffive  manner,  by  adding  the  intereft  to 
the  principal  at  the  time  of  the  firft  payment,  and  from 
the  fum  deducing  the  payment,  &c, 


aao  DISCOUNT. 


DISCOUNT 

IS  an  allowance  made  for  the  payment  of  any  fum  ot 
money,  before  it    becomes  due,   and  is  the  difference  be 
tween   thdt  fum,  due  fome   time  hence,    and  its  prefent 
worth. 

The  prefent  worth  of  any  Aim  or  debt,  due  fome    tims 
hence,  is  fuch  a  fum,  as,  if  put  to  intereli,  would  in  that 
time  and  at  the  rate  per  cent,  for  which  the  difcount  is  to 
made,  amount  to  the  fum  or  debt  then  due. 
RULE. 

As  the  amount  of  looi.  for  the  given  rate  and  time,  is 
to  lool.  :  fo  is  the  given  fum  or  debt  to  the  prefent 
worth. 

Subtract  the  prefent  worth  from  the  given  fum,  ani 
the  remainder  will  be  the  difccunt  required. 

Or,  As  the  amount  of  iool.  for  the  given  rate  and 
time,  is  to  the  intereli  of  iool.  for  that  time  :  fo  is  the 
given  fum  or  debt  to  the  difcount  required. 

Or,  In  Federal  Money*  divide  the  given  fum  by  the  a- 
mount  of  Dioo  for  the  given  time  and  rate  ;  point  off 
from  the  right  of  the  quotient  two  places  lefs  than  in  di- 
vifion  of  decimals,  for  the  prefent  worth. 

EXAMPLES. 
i.  What  is  the  difcount  of  J4^5   'os.     |  due   ^ 

{^Lf2  I  19      5CC.  J 

years  hence,  at  5^  per  cent,  per  annum  ? 
Intereft  of  iool.  per  annum =5    to 

2  years. 

1 1 

Add  100 

1 1 1 

As  £111  :  £u   ::  635!.   175.  :  63!.  os.  3|d.  difc.  Ans. 
Or, 

£      £       £   *     £     •-   d. 

As   in   :    100  ::  635    17  :  572   16  9^  prefent  worth. 
And  635!.  173.— 572!.  1 6s.  9jd.=6$l.  os.  2^d.  difcount^ 


BARTER. 

In  Federal  Money. 
D.      D.         D.     c.         D.    c.  m. 

As   ii  t    :    1 1   ::  21 19  50  :   210  04  o^rrdifcount.      Or, 

21 19-5X100 

As   in    :  100 ::   2119  50  : =01909  450. 

1 1 1 

9-g-m=prefent  worth  ;  and  21 19  5 — 1909  4595=210-0405 
^ilifcount,  as  before. 
2119-5 

Or,   -- =19-094595    j    and    1909-4095  =  preft-nt 

1 1 1 
worth  as  before. 

2.  What  is  the  prefent  worth  of  0350  payable  m  half 
a  year,  difcounting  at  6  per  cent,  per  annum  ? 

Ans.    0339  8oc.   5nv 

3.  What    is  the  prefent  worth  of  65!   due  15  months 
hence,  at  61   per  cent-  per  annum  ?         Ans.  6ol.  95.  3^d. 

4.  What  is  the   difcount  on  97!,    los.  due  January  22, 
this  being  Sept.  7,  reckoning  interest  at  5!   per  cent,  r 

Ans.    ii.  155    iid. 

5.  Bought  a  quantity  of  goods  for  1)250  ready  mon- 
ey, and  fold  them  for  Djco  payable  9    months  hence  ; 
What  was  the  gain  in  ready  money,  fuppoiing  difcount  to 
be  nude  at  6  per  cent.  ?  An?.  Dj7  8c    i£m. 


BARTER 

IS  the  exchanging  of  one  commodity  for  another,  and 
teaches  traders  to  proportion  their  quantities  without  lofs. 

CASE  I. 

When  the  quantity  of  one  commodity  is  given,  with  its  valus,  or 
that  of  its  integer )  that  /'/,  of  \lb.  \ciut  lyd  &c.  as  alfo 
the  value  of  the  integer  of  fome  other  commodity,  to  be  given 
Jor  it,  to  find  the  quantity  of  this  ;  or  having  the  quantity 
thereof  given,  to  find  the  rate  of  felling  it. 

RULE. 

Find  the  value  of  the  given  quantity  by  the  concifed 
method,  then  find  what  quantity  of  the  oilier,  at  the  rate 
propoied,  you  may  have  for  the  fame  money  :  Or,  if  the 

T  * 


BARTER. 

quantity  be  given,  find  from  thence  the  rate  of  felling  it. 
Or,  As  the  quantity  of  one  article  U  to  its  price,  fo,  t>;verfe/y9 
is  th*  quantity  of  the  ether  to  its  price.  Or,  as  the  price 
of  one  article  is  to  its  quantity  ;  fo  inverfely,  is  the  price, 
of  the  other  to  its  quantity. 

EXAMPLES. 

f .  How  much  tea,  at  95.  6d.  per  Ib.   mud  he  given  m- 
barter  for  156  gallons  of  wine,  at  izs.  3^d.  per  gallon  ? 
Gals. 


d.     Ib. 

As  114  :    i   ::  23010  :  201      i3fV;4  Ans. 

price.          quan.  price.        quan. 

Or,    As   i2s  3^d.  :    i56gal.  ::  93-   6.  :   2Oilb.  i 
anfwer,  as  before. 

2.  How  much   cloth,  at   155.   ftd.  per    yard,  muft  be 
given  for  5cwt.  $qrs..  i9lb.  of  fteel,  at  5  guineas  per  cwt  ?'.' 

Ans.  52yds.  3qrs    2n» 

3.  Suppofe   A   has  350  yards  of  linen,    at    250.  per 
yard,  which  he  would  truck  with  B  for  fugar,  at  JL>5  per. 
cwt.     How  much  will  the  iinsn  con^e  to  ? 

Ans.    lycwt.  2qrs. 

4.  A  has  broadcloths   at   1)44  per  piece,  and  B  has 
mace,  at  Di  42C.    pcilb    :     How  many  pounds  of  mace 
mill  B  give  A  for  35  pieces  of  cloth  ?      Ans.    tSo^lbs. 

5.  A  has  ylcvvt  of  fugar  at  12  cents  pei  In.   for  which 
Bgave  him   i^cwl.  c-f.  flour  :    What  was  the  flour  rated 
at  per  lb,.  },  Ans,  7c.  zm. 


BARTER.  223 

CASE    II. 

If  the  quantifies  of  (wo  commodities  be  given*  and  the  rate  of 
felling  th?m,  to  find*  in  cafe  of  inequality,^  hoiv  much  of  fome 
other  cwamodity  mujl  be  given. 

RULE. 

Find  the  feparate  value ;  of  the  two  given  commodities  ; 
fubtracl  the  it'fs  from  the  Alcatel,  and  the  diff  rw  ce  will 
be  the  amount  of  the  third  commodity,  whofe  quality 
and  rate  may  be  tafily  found. 

EXAMPLE?. 

1.  TVSM  merchants  b.irrer  ;  A  has  3ocwt.  of  cheefe,  at 
235.  6d.    per  cwt.    and  B  has  9  pieces  of   broadcloth,   at 
3!    155.  per  piece  t  Which  muU  leoeive  money,  and  how 
jariuch  ?  An?.   B  muit  pay  A    il.   IDS. 

2.  \  and  B.  would  barter  ;   A  has  150  bufheis  of  wheat* 
at  Di    250   per  hufhel*  for  which   B   giv.'s  65  bufheis   of 
b.uley,  worth  62\c   p-r  bufhel,  and  the  balance  in  oats  at 
37ic   perbufiiel  ;•  Wiidi  quantity  of  oats  miut  A  receive 
from  B  ?  AnSx   39 1|  buibels. 

CASE  III. 

Sometimes,  in  bartering*  one  commodity  is  rattd  above  the  ready 
money  price  ;  then,  to  find  the  bartering  price  of  the  other^ 

/*>'< 
As  the  ready  money  price  of  the  one,  is  to  its  bartering 

pric'.  ;  fo  is  that  ot  the  other,  to  its  b  -.rtering  price  :  Next, 
find  the  quantity  required,  according  to  either  the  barter- 
ing or  re^tdy  money  price 

EXAMPLES. 

i  A  has  iibbands  ^  2s.  pet  yard  ready  money  ;  but 
in  >  ;  ter  he  will  h  ive  2s.  3d.  ;  B  h-is  bioadcloth  at  323^ 
61  per  yard  readv  money  ;  at  whut  rate  muit  B  value  his 
cloth  per  yard  to  be  equivalent  to  A's  bartering  price, 
and  h«>w  many  yards  cf  libbaiid,  at  2s.  3d.  per  yard,  mud 
then  be  given  by  A  for  4:^8  jurds  of  B's  broadcloth  ? 

Ans  13's  broadcloth,  ac  £i  i6s.  6gd.  per  yd.;  7950. 
yds  ribband. 

i.  \  and  13  barter ;  A  has  150  gallons  of  brandy,  at 
Di  37.^  P^r  gallon  ready  m-)  ey,  out  in  barter  he  will 
h  iv.-  O  t  5<DC.  ;  B  ha>  linen  i»  4|C-  per  yard  ready  mbr.ey  ;. 
bow  mu.t  B  feii  hi>  linen  per  yard  in  propoition  to  x-i's 


224  LOSS  AND  GAIN. 

bartering   price,  and  how  miny  yards  are  equal  to  A's 
brandy  ? 

Ans.  Barter  price  is  48c.  and  he  rnuft  give  A  468! yds. 
3  P  and  QJ^rter  :  Phas  Irifli  linen,  at  6oc.  per  yard, 
but  in  barter  he  will  have  64C.  Q^  delivers  him  broul- 
cloih  at  D6  per  yard,  worth  only  D$  5oc.  per  yard:  Pray 
which  had  the  advantage  in  barter,  and  how  much  linen 
docs  P  give  Q  for  148  yard^,  of  broadcloth  ? 

c.       c,      D    c       D  c. 

As  60  :  64  ::  5  50  :  5  86|  ;  therefore,  Q^  by  fell- 
Ing  at  D6,  has  the  advantage.  Then, 

D      yds.       c.         yds.  qrs. 
As  6  :   148  ::  64  :   1387   2  linen,  Ans. 

4.  A  has  200  yards  of  linen,  at  is.  6d.  ready  money 
per  yard,  which  he  barters  with  B,  at  is.  pd.  per  yard, 
taking  buttons  at  7-J-d.  per  grofs,  which  are  worth  but  6d.  : 
How  many  grofs  of  buttons  will  pay  for  the  linen,  who 
gets  the  bed  bargain,  and  by  how  much,  both  in  the 
•whole,  and  per  cent.  ? 

yd.     d.       yds.         d.  d.  grofs.       d.     grofs. 

As   i   :   21.::   200  :  4200.     As  7^  :   i   ::  4200  :  560 

yd.     d.       yds.  £ 

As   i   :    1 8  ::  200   15  value  of  A's  linen. 
gr,     d..    gr.          £ 

As   i   :  6  ::   560  :   14  value  of  B's  goods.      So  that  B 
gains   il.  of  A. 

£    £    £ 

As   14  :    i   :   ico  :   7!.  2s.  icd.   per  cent, 

LOSS  AND  GAIN 

IS  an  excellent  rule,  by  which  merchants  and  traders  dif- 
cover  their  profit,  or  Iofs  per  cent,  or  by  the  grofs  :  It  alfo 
inllrucls  them  to  raife  or  fall  the  price  of  their  goods,  fo  as 
to  gain  or  lofe  fo  much  per  cent-  &c. 

CASE  1. 
To  know  what  is  gained  or  hft  per  cent. 

RULE 

Firft  fee  what  the  giin  or  iofs  is,  by  fubtraclion  ;  then, 
as  the  price  it  co/t,  i;;  to  the  gain  or  iofs  :  fo  is  lOoL  to  the 
gam  or  lois  per  cent.  &c. 


LOSS  AND  GAIN.  225 

Or,  mfeatfj/  mon':y>  annex  two  cyphers  to  the  gain  or 
loft,  and  divide  by  the  coft  for  the  gain  or  lofs  per  cent. 

EXAMPLES. 

i.  If  I  buy  ferge  at  900.  per  yard,  and  fell  it  again  at 
D  [  2c.  per  yard  :  What  do  I  gain  per  cent,  or  in  laying 
out  D  i  oo  ? 

c.       c.         P.       D. 

Sold  for  DrO2         As  90  :   12  ::   100  :   13^  per  cent* 
Cofl  -90  gaip,  Ans. 


Gain  -12  per  yard. 

I2'00 

Or,   ro2  —  '9O='i2=gain  per  yard  j  and    -  ^133-  per 

*9 

cent,  gain,  as  before. 

N.  B.     The  firft  queftions  in  the  feveral  cafes  ferve  to 
elucidate  each  other. 

2.  If  I  buy  ferge   at  Di   20.  per  yard,   and  fell  it  a- 
gain  at  90  c.  :  What  do  I   Icfe  per  cent,   or  in  laying  out 
Dioo? 

Coft     Di-cz  D.         c.       D.        D.  c.  m. 

Sold  for    '90     As   102   :    12  :   100  :    11    76  5     per 

—  .—  cent,  lofs,    Ans. 

Lofs         MI 

I2'CO 

Or,  --  =  1  1  2  1  5  per  cent,  lofs,  Ans.  as  before. 
i  02 

3.  If  I  buy  a  cwt.  of  tobacco  for  9!.  6s.   8d.  and   fell 
rt  again  at  is.  lod.  per  Ib.  do   I   gain  or  lofe,  and  what 
per  cent.  ?  £     s.    d. 

Sold  for   »o     5     4 
Coft  968 

0   1  3     8  gained  in  the  grofs, 

£    s- 

t&  I  T  2  I  1  1     4         value  at  2s.  per  Ib. 
o   1  8     8  value  at  2d.  per  Ib. 
10     c     4  value  at  is.  lod.  per  Ib. 

£«••».*£*•      £     £ 

As  9  6  8  :  18  8  ::  IOQ  :  10  Ans.  10  per  cent,  gain., 


22<S  LOSS  AND  GAIN. 

4.  A  draper  bought  60  yards  of  cloth  at  D^  joe. 
per  yard,  and  38  yards  of  cloth  at  D2  500.  per  yard, 
and  fold  them,  at  I>4  250.  per  yard  :  Did  he  gain  or 
lofe,  and  what  per  cent.  ? 

60  yards  at     D4  500.  per  yard     =      D2"o 
38  yards  at         2  50      per  yard     =  95 


98  yards  cod 

which  fubtraft  from  98  yards  at  D4  250.  = 

Gained  in  the  grofs     =          51*52 
D.      D.  c.         D,        5150-00      D.  c. 
Then,  as  365   :  51  -50  ::  100  : =14-11    gain 

3<5> 
per  cent.  Ans. 

5.  Bought  fugar  at  6^d.  per  Ib.  and  fold  it  at  2!.  35. 
9d.  per  cwt. :  What  was  the  gain  or  lofs  per  cent,  ? 

Ib.     d.        Ib.       £  s.  d. 
As   i   :  6^  ::   112   :  3   o  8 
Prime  coft  £3     o     8  per  cwt. 
Sold  at  239  per  cwt. 

Loft    £o  1 6  ii  in  the  whole. 
£  s.  d     £    s.          £         £     s.    d. 
As  3  o  8:  16   n  ::   100  :  27   17   8^  lofs  per  cent.  Ans, 

6.  At  45.  6d.  in  the  pound  proat,  how  much  per  cent.  ? 

£      s.  d.       £        £    s. 
As   i   :  4  6  ::   100  :  22   10  Ans. 

5.  If  I  buy  candles  at  is.  6d.  per  Ib.  and  fell  them  a- 
gain  at  23.  per  Ib.  and  allow  3  months  for  payment,  what 
do  1  gain  per  cent.  ? 

d.       d.        £        £    s.  d. 
As  18  :   24  ::   ico  :    133  6  8   ; 

Mo.  £     Mo.    £ 
Then,  by  Difcount,  As   12  :  6  ::  3  :   i    ics. 

£      s.         £  s.  £    s.  d.      £  s.    d. 

Then,  As  IOF  10  :  i  10  ::  133  6  8  :  i  10  4f 
which  taken  from  133!.  6s.  8d.  leaves  131!.  ys.  jjdi 
therefore,  Ans.  3,1!.  75.  3£d. 


LOSS  AND  GAIN.  227 

6.  If  I  buy  cloth  at  135.  per  yard,  on  8  months  credit, 
and  fell  it  again  at  12s.  ready  money,  do  I  gain  or  lofe, 
and  what  per  cent  ? 

Mo    £     Mo.    £  £          5.         £         s.  d. 

As  F2  :  6  ::   8  :   4  As    104  :    13  ::    100  :   12  6  ; 

So  that  1  35.  on   8  months  credit,  at  6  per  cent,  is  equal 
to  I2s.  6d.  ready  money  ;  then, 

Prime  eoft       izs.  6d.  ready  money. 
Sold  at  1  2     o      ready  money. 

Loft  6      in  the  yard. 

As   I2s.  6d.  :  6d.  ::  £100  :  £4      Ans.  4!.  percent. 

CASE     II. 

To  know  how  a  covtwodity  muft  be  fold,  to  gain  or   lofe  fo  much 
per  cent. 

RULE.  As  icol.  is  to  the  price  ;  fo  is  icol.  with  the 
profit  added,  or  lofs  fubtracled,  to  the  gaining  or  lofing 
price. 

Or,  In  Federal  Money,  multiply  ico  dollars  added  to  the 
gain,  or  lefs  by  the  lois  per  cent  by  the  cod  ;  and  point- 
ing off  the  two  right  hand  figures  of  the  product  gives  the 
anfwer. 

EXAMPLES, 

1  .  If  I  buy  a  quantity  of  ferge,  at  9Oc.  per  yard  :  How 
muft  I  fell  it  per  yard  to  gain  \  3  J  per  cent.  ? 

D.      D.     c.          c.     D.c. 
As   100  :   113  33^  ::  90  :   i   2    Ans. 
Or,  Dii3  33-Jc.  X9oc  =Dio2  ;  and  pointing  off  two 
right  hand  pLcess  Dro2,  Ans  as  before. 

2  If  a  barrel  of  powder  coft  4!.  how  irmft  it  be  fold 
to  lofe    10    er  cent.  ? 


As   100  :  4  ::  90  Or  thus  : 

4  9° 

100)360(3 

300  £3)60 

—  20 

60-  - 

2O  S,  I2|OO 

100)1200(12  Ans.  £3  12 
1  200 


X2S  LOP?    INt)  GAIN. 

In  CM.O i.r\e  up  the  amount  of  goods  bought)  imported  or 
exported  :  to  the  piime  coftrt  fuch  goods  we  mud  add  all 
the  charges  upon  them,  in  order  to  fix  the  price  they  (land 
us  in. 

3.  tSuppofe  I  import  from  France,  12  bales  of  cloth, 
containing  i  o  pieces  tach,  which,  with  the  charges  there, 
amounted  to  Djflc  :  I  pay  duty  here  y2c.  per  piece,  for 
freight  Di  2,  and  porterage  Di  250.  ;  What  does  it  ftand 
me  in  per  piece,  and  how  mull  I  fell  it  per  piece  to  gain 
Dio  per  cent  ? 

Firft  coft 

12  bales.  Duty 

I  o  Freight 

—  Porterage 

1 20  pieces. 

Whole  coft     483-65 

Pieces.         D.  c.       Piece.      D.c. 
As   120     :     48365    ::     i      :     4-03  coft  per  piece. 

D.     per  ct.  D.c.  c.  m. 

Again,  as   100  :    10  ::  4-03     ;     40  3  gain. 
40  3 

D443  3  "the  price  at  which  it  muft 
be  fold  per  piece. 

4  Bought  cloth,  at  Di  £oc.  per  yard,  which  not  prov- 
ing fo  good  as  I  expected,  I  am  content  to  lofe  17^  per 
cent,  by  it  :  How  muft  I  fell  it  per  yard  ? 

Ans.  D2  6c.  2jm. 

5.  If  i2olb.  of  fteel  coft  7!.  how  muft  I  fell  it  per  Ib. 
to  gain  15^1.  per  cent.  ? 

As  I2olb.  :  £7  ::   lib.  :    is.   2d. 
As  £too  :    is.  2d.   ::  £115-^  :  is.  4d.   per  Ib.  Ans. 

6.  Bought  fifti  in  Newburyport,  at  IDS.  per  quintal,  and 
fold  it  ac  Philadelphia,  at  175   6d.  per  quintal  ;    now  al- 
lowing the  charges  at  an  average,  or  one  with  another,  to 
be  2s.  6d,   per  quintal,    and  coniidering  I  muft  lofe  20!. 
per  cent,  by  remitting  my  money  home  j  what  do  I  gain 
per  cent,  ? 


LOSS  AND  GAIN*  229 

Selling  price    17  6  Philadelphia  currency,  per  quintal. 
Charges          2   6  ditto. 

150  ditto. 

£•      *•      £      «• 

As  100  :   15    ::    So    :    1 2  New  England  currency. 
Sold  at          i  2s     per  quintal. 
Bought  at    fos    per  quintal. 

Gained  2s.    per  quintal, 

s.       s.         £.         £. 

As  10  :  2  ::     ico    :    20  percent    gained,  Ans. 

7.   Bought  50  gallons  of  brandy,  at  750.  per  gallon,  but 

by  accident,  10  gallons  leaked  out  :   At  what   ratt  mu.t  I 

fell  the  remainder  per  gallon,  to  gain  upon  the  whole  prime 

colt»  at  the  rate  of  10  per  cem.  ?  Ans.  Di    $c.   i^m. 

CASE      III. 

When  thtrc  is  gain  or  lofs  per  cent*  to  know  what  the  commod- 
ity cofl' 

RULE.  As  lool.  with  the  gain  per  cent,  added,  or  lofs 
per  cent,  fubtraded,  i;>  to  the  price  ;  lo  is  lool.  to  the 
prime  coit. 

Or,  In  Federal  Money,  divide  the  price  with  two  cyphers 
annexed  by  Dioo  added  to  the  gam,  or  lels  by  the  iofs, 
per  cent,  for  the  anfwtr 

EXAMPLES. 

1.  If  i  yard  of  cloth   be  i^d,  at  Di  2C.  and  there  is 
gained  13-}  per  cent,   wh  ,t  diu  the  yard  coil  ? 

D.c.       D.         c 

As  Dioo+«35  :   i   2  ::   100  :  90  prime  coft,  Anfwer. 
102*00 

Or,  ='9   Ans.  as  before. 

^333T 

2.  If  12  yards  of   cloth  are  foM  at  i  js.  per  yard,  and 
there  is  7!      los   lofs  per  cent,  in  the  fait  :     What  is  the 
prime  cod  of  the  whole  ? 

Yd.        s.  yds.         £ 

As   i      :     15     ::      12     :     9 

£    *•        £        £          £  s.    d. 

As  92   10     :     9     ::     100     :     9  14  7  Ans. 
IT 


LOSS  AND  GAIN. 

3  If  i  Q£  cwt.  fugsr  be  fold  at  D 14  500.  per  cwt.  and  I 
gain  DIJ  per  cent  :  What  did  it  coil  per  cwt.  ? 

9.  B.C.  D.  D 

As  115     :     14*50     ::     100     :     12-608   Ans, 

CASE    IV. 
If  bv  wares  fold  at  (itch   a  rate,  there  is  fo  much  gained  or  loft 

per  cent,  to  know  what  ivculd  be  gained  or  loft  per  cent,  if 

fold  at    another  rate. 

RULE, 

As  the  firft  price  is  to  lool  with  the  profit  percent, 
added,  or  lofs  per  cent  fubtracled  ;  to  is  the  other  price, 
to  vh^  gain  or  lofs  per  cent,  at  the  other  rate. 

N.  B,  if  your  anfwer  exceed  ioov  the  excels  is  your 
gain  per  cent.  ;  but  if  it  be  lefs  than  100,  the  deficiency  is 
your  iois  per  cent. 

EXAMPLES. 

1.  If  cloth  fold  at  Di    2c    per  yard,   be  13^  profit  per 
cent  v-hat  gain  or  lofs  per  cent,  ihail  1  have,  it  I  ieli  the 
fame  at  9 oc.  per  yard  ? 

D  c.  D.  c.          D. 

^As   i   2     :     113!     ::     90     :     r««o 
And   100 — 1 00=0  Ans.  I  neither  gain  nor  lofe. 

2.  If  cloth,  fold  at  4  .  per  yard,  be  lol.  per  cent,  profit : 
What  fhall  I  gain  or  ioie  per  cent,  if   fold  at  35    6d.  per 
yard  ? 

8.  £  S.  d. 

As  4     :     no     :     36 
12  12 

48  42  Then,  icol — v6£l.~3|L 

no 

42)4620(96^  Ans.  I  loft  3^1    per  cent,  by 
432  the  lad  iale. 

300 
288 


EQUATION  OF  PAYMENTS.  231 

3.  If  I  fell  a  gallon  of  wine  for  Di  500.  and  thereby 
lof  i  2  per  cent.  :  Whdt  fiiall  I  gain  or  lofe  per  cent,  if 
I  fell  4  gallons  of  the  fame  wine  for  D6  750.  ? 

D      c.        DC.       c. 
As  6  :  88  :;  6  75  :  99  And  100 — 99=1  per  cent,  lufs. 


EQUATION  OF  PAYMENTS 

IS  the  finding  a  time  to  pay  at  once  feveral  debts  due 
at  different  times,  fo  that  no  lofs  (hall  be  iuftained  by  ei- 
ther party. 

RULE     I.* 

Multiply  each  payment  by  the  time  at  which  it  is  due  ; 
then  divide  the  fum  of  the  produces  by  the  fum  of  the  pay- 
ments and  the  quotient  will  be  the  equated  time,  or  that 
required. 

EXAMPLES. 

i.  A  owes  B  0380  to  be  paid  as  follows,  viz.  Dioo 
in  6  months,  Di2o  in  7  months,  and  Di6o  in  10  months  i 
What  is  the  equated  time  for  the  payment  of  the  whole 
debt  ? 

i  ocX  6=  600 
I20X  7=  840 
160X10=1600 

100+1-20+160=380)3040(8  months,  Ans. 
3040 


2.  A  owes  B  104!.  155.  to  be  paid  in  4^  months,  i6iL 
to  be  paid  in  3^  months,  and  152!.  55.  to  be  paid  in  5 
months  :  What  is  the  equated  time  for  the  payment  of 
the  whole  ?  Ans.  4  months  and  8  days. 


*  This  rule  is  founded  upon  the  fiippofUion  that  the  fum  of  the 
interefts  of  the  feveral  debts,  which  are  payable  before  the  equated 
time,  from  their  terms  to  that  time,  ought  to  be  equal  to  the  fum  of 
the  interc.ts  of  the  debts  payable  after  the  equated  time,  from  that 
time  to  their  terms. 


EQUATION  OF  PAYMENTS. 

3.  There  is  owing  to  a  merchant  998!.  to  be  paid,  178!, 
ready  money,  200).  at  3  months,  and  320!.  in  8  months  ; 

1  demand   the  indifferent   time  for   the  payment  of  the 
whole  ?  Ans.  4^  months. 

4.  The  fum  of   I)  (64  i6c.    6m.  is  to  be   paid,  ^   in  6 
months,  y  in  8  months  and^-  in  12  months  :  What  is  the 
mean  time  for  the  payment  of  the  whole  ? 

Ans.  7y  months. 

RULE   II. 

See  by  Rule  i,  at  xvhat  time  the  firft  man,  mentioned, 
eughr  to  pay  in  his  whole  money  ;  then  as  his  money  ii  to 
his  time,  fo  is  the  other's  money,  to  his  time,  inverfely, 
which  when  found,  muft  be  added  to,  or  fubtrafted  from, 
the  time  at  which  the  iecond  ought  to  have  paid  in  his 
money,  as  the  cafe  may  require,  and  the  fum,  or  remain- 
der, will  be  tbe  time  of  the  fecond's  payment. 

EXAMPLES. 

1.  A  is   indebted  to  B   D^c,  to  be   paid,  D^o   at  4 
months,  and  DIQO  at  8  months  :  B  owes  A  0250,  to  be 
paid  at  10  months  :     It  is  agreed  between   them  that  A 
ihall  make  prefent  pay  of  his  whole  debt,  and  that  B  (hall 
pay  his  fo  much  the  fooner,  as  to  balance  that  favour  :  I 
demand  the  time  at  which  B  muftpay  the  Dajo,  reckon- 
lag  fimple  intereft. 

50X4=200 
100X8=800 

$o+ioc=i5|o)ioolc(6f  months,  A's  equated  time, 
90 

10 

D*         mo.         D.      ma. 

As   150     :     6|    ::    290    :    4 

mo.  mo.  mo. 
Then,  ic — 4  =  6  time  of  B's  payment. 

2.  A  merchant  has  Di2oo  due  to  him,  to  be  paid  £  at 

2  months,  ^  at  3  months,  and  the  reft  at  6  months  ;  but 
the  debtor  agrees  to  pay  ^  down  :    How  long  may  the 
debtor  detain  the  other  half,  fo  that  neither  party  may  fuf- 
tairj  lofs  ? 


POLICIES  OF  INSURANCE.  233 

mo.     mo. 


4X6=3 


Equated 

Now,  as  \  was  paid  4!-  months  before  it  was  due,  it  is 
reafonable  that  he  ihould  detain  the  other  g  4^  months 
after  it  became  due,  which,  added,  gives  8§  months,  the 
true  time  for  the  fecond  payment. 


POLICIES  OF  INSURANCE. 

INSURANCE  is  a  fecurity,  or  aflurance,  by  mean  of 
a  writ  called  a  Policy,  to  indemnify  the  infured  of  fach. 
lo/Tes  as  (hall  be  fpecificd  in  the  policy  fubfcribed  by  the 
iniurer,  or  infurers,  by  whicli  the  under  writers  oblige 
themfelves  to  make  good  and  effectual  the  property  in- 
fured, in  consideration  of  a  certain  premium  at  a.  ftipula- 
ted  rare  per  cent,  (which  varies  according  to  the  riikj  to 
be  immediately  paid  down,  or  otherwife  fecured  accord- 
ing to  the  tenor  of  the  agreement. 

The  average  lofs  is  10  per  cent.  ;  that  is,  if  the  infur- 
ed fufter  any  dam.'ge  or  lofs,  not  exceeding  10  per  cent, 
he  bears  it  himfeif,  and  the  infurers  are  free. 

A  policy  ihould  be  taken  out  for  A.  ium  fufficient  to 
cover  the  principal  and  premium,  and  the  buiinefs  of  this 
rule  is,  in  general,  to  calculate  what  that  fum  fhouid  be. 


CASE    L 

When  the  premium^  at  a  certain  rate  per  cent,  for  hairing  a 
Jum,  if  requirsdy  the  operation  is  ths  faun  as  in  intertftt  or 
coimmjjion. 

EXAMPLES. 

i.  What  is  the  premium  upon  537!.  153,  9d.  ay  6J  per 
cent-  ? 

U  2 


234  POLICIES  OF  INSURANCE. 

£      s.    d. 

.537     «5     9 
fcj 


3226     14     6 

|=       268        17     10-3- 

34195     12     4i 
20 

19(12 

12 


1(94  Ans.  34!.  195.  ijd.  nearly. 

:.  What  is  the  premium  upon  0375,  at  7^  per  cent,  ? 


•075 


Ans.  D2S-I25 

CASE    II. 

To  fnd  the  fumfor  which  a  policy  Jhould  Is  taken  out  to  covtr 
a  given  ftim. 

RULE. 

Take  the  premium  from  tool,  (or  in  federal 'money  D  100) 
and  fay,  As  the  remainder  is  to  100,  fo  is  the  ium  adven- 
tured to  the  policy.*  Or, 

*  Now  it  is  plain,  that  if  I  want  to  recover  921. 1  muft  in  this  cafe, 
ii.fure  upon  1001.  :  therefore  to  recover  7591.  I  muft  infure  upon 
tf  251. ;  for  when  8  per  cent,  for  premium  is  deducted,  I  ihall  have 
"  ~.fj\.  remaining  nett. 

For  ^825=fuin  inuiredupon  at  8  per  cent, 
6t>"=premium  to  be  d 

759— the  firft  outfet. 


POLICIES  OF  INSURANCE.  23$ 

In  decimals,  take  the  premium  from  i  oo,  annex  two 
cyphers  to  the  Aim  to  be  covered,  and  divide  by  the  re- 
mainder for  the  policy. 

EXAMPLES. 

i.  It  is  required  to  cover  759!.   premium  8  per  cent.  : 
For  what  fum  muft  the  policy  be  taken  ? 
100 
8 

92     :     100     ::     759 
100 

97)7^900(825!.  Ans. 
736_ 

230 


460 
460 
75900 
Or,    --  -  =825!.  Ans.  as  before. 

92 

2.  A  merchant  fent  a  vefTel  and  cargo  to  fea,  valued  at 
05760  :  What  fum  muft  the  policy   be  taken  out  for,  to 
cover  this  property,  premium    1  9^  per  cent.  ? 
100 
195 

-          D.  D.  D.     c. 

80*5     :      100     ::     5760     :     7155-28  —  Ans. 

576000 
Or,  -  -----  =07155-280.  —  Ans.  as  before.  4 

80-5 

CASE     III. 

When  a  policy  is  taksn  out  for  a  certain  fum,  la  order  to  cov- 
er a  given  fum. 

RULE. 

To  find  the  premium,  fay,  As  the  policy  is  to  the  cov- 
ered fum  ;  to  is  ^'oo  (-,r  Dtco)  to  a  fourth  number, 
which,  being  taken  from  100,  will  leave  the  premium, 


236  POLICIES  OF  INSURANCE. 

Or,  In  decimals,  divide  the  fum  covered,  with  two  cy- 
phers annexed,  by  the  policy  ;  fubtract  the  quotient  from 
loo,  and  the  remainder  is  the  premium. 
EXAMPLES. 

1,  If  a  policy  be  taken  out  for    1250!.  to  cover  500!. 
what  is  the  premium  per  cent.  ? 

1250     :     500     ::      100 

ICO 

1250)50000(40.  and   icol. — 40=60!.  Ans. 
50000 

Or, —40,  &c.  as  before. 

1250 

2.  If  a    policy  be   taken  out  for  0781*25,  to  cover 
D6^;   :   Required  the  premium  per  cent. 

D.  D.         D.      D.   c. 

As  781-25  :  625  ::    100  :  87  50         And   100 — 87-5= 
125,  or  I2g  percent,  premium,  Ans. 
62500 

Or, =87*5,  &c.  as  before. 

781-25 

CASE    IV. 

When  the  policy  for  covering  any  fum  and  the  premium  p-:r 
cent,  are  given,  to  find  the  fam  to  be  covered. 

RULE. 

Deduct  the  premium  per  cent,  from  ico,  and  fay,  As 
ico  is  to  the  remainder,  fo  is  the  policy  to  the  fum  re- 
quired to  be  covered.  Or, 

In  decimals^  multiply  the  policy  by  the  remainder  .found 
as  before,  and  point  off  two  right  hind  places  in  the  pro- 
duct for  the  anfwer. 

EXAMPLES. 

i.  If  a  policy  be  taken  out  for  £1250  at  60  per  cent, 
what  is  r.he  adventure,  or  fum  to  be  covered  : 
100 
60 

ico    :     40      ::      1250 

40 

100)50000(500!.  Ans, 


COMPOUND  INTEREST.  23* 

Or,  1250X100 — 6c= 50000,  anJ,  pointing  off  two 
places,  500*00  Anfwcr,  as  h-etor^. 

2.  If  a  policy  be  taken  out  for  DySi  250.  at  12^ 
per  cent,  required  the  fum  covered  ? 

781-25X100 — 12^ 
As  100  :  IOQ — 13 j  ::  781*25  : ' «*• 

ICO 

D5^5  Anfwer. 

Or,  78i<2.5X»oc-*-i2  5=62500  ;  and  62500  Ans.  ag 
fcefore. 


COMPOUND  INTEREST 

IS  that  which  anfes  from  the  intereft  being  added  t« 
the  principal  .-.nd  (continuing  in  the  h-.inds  of  the  borrow- 
er) becoming  part  of  the  pricipal,  at  the  end  of  each  ftated 
time  of  pay  menu. 

METHOD    I. 
RULE.* 

Find  the  amount  of  the  given  principal,  for  the  time  of 
the  firft  payment,  by  Simple  Intefeft  :  next,  find  the  in- 
tereft  of  that  fum,  or  principal,  and  add  it  as  before,  and 
thus  proceed  for  any  number  of  years,  ftiil  accounting  the 
laft  amount  as  the  principal  for  the  next  payment  The 
given  principal  being  fubtradled  from  the  lad  amount,  the 
remainder  will  be  the  compound  intereft. 

\nfederal  money ,  multiply  the  principal  by  the  rate  for 
the  firft  time  of  payment,  fetting  the  product  two  places 
m  >re  to  the  right  than  the  multiplicand,  and  the  decimal 
point  in  the  product  under  that  in  the  multiplicand  ;  then 
find  the  amount,  and  proceed  as  above. 

*  It  may  be  obferved  that  all  computations,  relating  to  Compound 
Intereft,  are  founded  upon  a  feries  of  terms,  increafing  in  Geometri- 
cal Progreilum,  wherein  the  number  of  years  affigns  the  index  of  the 
laft  and  higheft  term  :  Therefore,  as  one  pound  is  to  the  amount  of 
one  pound,  for  any  given  time,  fo  is  any  propofed  principal,  or  iumt 
to  its  amount  for  the  fame  time. 


23  8  COMPOUND  INTEREST. 

NOTE.  It  is  not  ufually  neceffary  to  carry  the  work  be- 
yond mills  ;  therefore,  when  the  figure  next  beyond  mills5, 
at  the  right,  exceeds  5  increafe  the  r.  umber  of  milis  •  ; 
xvhen  it  does  not  exceed  5  it  nnuy  be  omitted  The  re- 
fult  will  be  exact  enough  for  common  purpofes. 

EXAMPLES. 


i.  What  will    |f^°    f    amount  to  in  5  years,  at  6 


per  cent,  per  annum 
Principal 
Rate  of  interest 


2^,80 

2.0 


i6|co 

Principal  for  the  fir  (I  year        £480     d 
Inure  ft  of  ditto  28   »6 

Principal  for  the  fecond  year    508   16 

6 


30(52   16 
20 

io|56 

12 


6|73 

4 


Principal  for  the  fecond  year    £508     16 
Intereft  for  ditto  30     10 

Principal  for  the  third  year          539       6 


COMPOUND  INTEREST.  239 

Principal  for  the  3d  year         £5  «9       6     6  J 
Intereft  for  ditto  3  ?        7      2% 

Principal  for  the  4th  year         5  /  1      1  3     8| 

_  _  6_ 
34(30       2.  44- 
20 

6|C2 
12 

©J28 

4 


Piincipal  for  the  4th  year        ^*57«     13 
Irncieu  fordo.  34       6 


Principal  for  the  5th  year  60  j     19     9 

6 


Principal  for  the  5th  year 
Interert  for  ditto 

Amount  for    5  years 
Subtract  the  firft  principal 


Compound  intere ft  for  5  years         162       6  n 

In   Federal  Money  thus  : 
Principal  for  the  firft  year       D « 600* 
Kate  of   interelt  6 

Intereft   firft  year  96-00 

Amount  ift  and  principal  2dyear  ' 


Carried  over, 


COMPOUND  INTEREST. 

Brought  over  1696* 

6 

Intereft    2d  year     .  101-76 

Amotmt  2d  year,  principal  3d  *  797*76 

6 


Intereft  3d  year  107-^656 


Amount  3d  year,  principal  4th         1905-6256 

6 


Intereft  <jth  year  !I4'3375$6 

Amount  4th  year,  principal  5th      2019-963:36 

6 

Intereft  jth  year  12 1  19778816 

Amount  for  5  years  2141-16092416 

Subtract  fir  it  principal  1600- 


Compound  intereft  for  5  years    =      541-1609241$ 

Or  thus  : 

1). 

Firft  principal  1600* 

6 

Intereft  96-00 

Second  principal       1 696* 
6 

Inteieft  101  76 


Thhd  principal        *79?  7^ 
6 

Intereft  ^07  ^66 

Fourth  principal      1905 -62-$ 


COMPOUND  INTEREST.  241 

Fourth  principal     D 1905 -626 

6 


Intereft  114-338 

Fifth  principal  2019^64 

6 


Intereft  121-198 

Amount         2  141* '62 
i  ft  principal  1600* 

(fore. 

Compound  intereft  541*162  nearly,  as  be- 

2.  What  is  the  compound  intereft  of  Dy4O  for  6  years, 
4  per  cent,  per  annum  ?  Ans.  1)196  33C.  im. 


3.  What  will    4  Q^Q  f  amount  to  in  a  year,  at  2  per 

cent,  a  month  ?  A       C^^o    4s     5^. 

;<lU634  i2c,  6m. 

METHOD  II. 
When  tie  rate  is  at  5  per  cent,  per  annum. 

1.  Divide  the  principal  by  20,  and  this  quotient,  added 
to  the  principal,  will  be  the  amount  for  the  iirft  year,  and 
the  principal  for  the  fecond. 

2.  In   like  manner  find  the  amount  for  every  fucceed- 
ing  year. 

When  the  rate  is  at  6  per  cent,  per  annum. 

\ .  Divide  the  principal  by  20,  and  that  quotient  by  5  : 
thefe  quotients,  added  to  the  principal,  will  be  the  amount 
for  the  firft  year,  and  the  principal  for  the  fecond. 

2.  In  like  manner  obtain  the  amount  for  every  fucceed- 
ing  year. 

W 


COMPOUND  INTEREST 


i.  What 
of  480!.  at 
annum,  for 

20)480 

5)   24 
4  16 


EXAMPLES. 

is  the   amount       2.  Of  the  fame  Aim  at  5 
6   per  cent,  per  per  cent,  for  3  years  ? 
5  years  ?  20)480 

24 

-  ~ 
20)504      amount  of  ift  yr. 

amo.  of  i  ft  year.        - 
9^  20)529 

9 


4     ditto  of  2d. 
27     9     2\ 


20)539     6  64  ditto  of  zd.  20)555  13  2%  ditto  of  3d. 

5)   26  19  3i  26  15  v| 

5     7  io£  

20)583  8  10     do.  of  4th. 

20)571    13  8^  dittoof  3d.  29  3  5£ 

5)  28   ii  8| 

5  H  4 


20)605   19 

5)  30     5 
6     i 


£612   12     3^:  do.  of  5th, 
—  Ans. 

8|  ditto  of  4th. 

i^  NOTE.  The  fame  may  be 

2£  done  in  Federal  Money,  but 

the  firft  method  is  generally 
o.5th, Ans.  more  eafy. 


COMPOUND  INTEREST  BT  DECIMALS. 


A  TABLE  of  the  amount  of  £\  or  Di,  at  \  per  cent, 
per  month,  as  praclifed  at  the  Banks. 


Mon. 

£  orL>. 
dec   pts. 

Mon. 

j£orD. 
dec.  pts. 

Mon. 

£orD. 
dec.  pts. 

1 

2 

3 

4 

1-005 

I-GI 
1-015 
I«Q2 

i 

i 

1-025 

1-03 
1035 
1-04 

9 

10 
12 

1-045 
1-05 
1-055 

i  -06 

BY  DECIMALS. 


A  TABLE  of  the  Amount  of  £i  or    Di,  from  i  Day 
to  3 1  Days,  at  6  per  cent,  per  annum. 


Days. 

£or  D. 

dec.  pts. 

Divs 

£  orD. 

dec.  pts. 

Days 

£or  D. 

dec.  pts. 

i 

00016 

12 

1-00197 

22 

00361 

2 

fOOO3  2 

13 

1-00213 

23 

•00378 

3 

•OOC4Q 

14 

1-0023 

24 

•00394 

4 

•00065 

15 

1*00246 

25 

•0041 

5 

•OOO82 

16 

1-00263 

26 

•00427 

6 

•OOOpS 

<7 

1*00279 

27 

•00443 

7 

•OOII5 

18 

1-00295 

28 

•0046 

8 

•0013  I 

19 

1*00312 

29 

•00476 

9 

•OOI47 

20 

1-00328 

30 

•00493 

10 

•00164 

21 

i  00345 

3^ 

00509 

1  1 

'OOl8o 

CASE    I. 

When  the  principal,  the  rate  of  inter ejl,  and  timc^  arc  givst:t  i'j 
i  jind  cither  the  amount  or  inter  eft. 

RULE. 

1.  Find  the  amount  of  £i  or  Di   for  one  year  at  the 
given  rate  per  cent. 

2.  Involve  th?  amount,   thus  found,  to  fuch  power,  us 
is  denoted  by  the  number  of  years  ;  or,  in  Table  1,  at  the 
end  of   Annuities,  under  the  rate  and  againft  the  given 
number  of  years,  you  will   find  the  power. 

3.  Multiply  this  power  by  the  principal,  or  given  fern, 
and  the  produd  will  be  the  amount  required,  from  which, 
if  you  fubtraft  the   principal,  the  remainder  will  be    the 
inter«,il. 


EXAMPLES. 

r.  What  is  the  Compound  intereft  of  6ool,  for  4  years, 
at  6  per  cent,  per  annum  ? 


244  COMPOUND  INTEREST 

c     f  amount  of  il.  for  i  year* 
i  -co=  <  f  * 

£      at  6  per  cent,  per  ann. 

Multiply  by  i  x  6 


Multiply  by 


=4t     power. 
Multiply  by  6oc=principal. 


75  7*486  i76oo=;amount. 
Subtract          6co 


157486176=^157    95.    8  Id.    intereft    re, 
quired. 

By    TABLE     I. 

Tabular  amount  of  il.  for  4  years,!  /-       /- 

at  6  percent,  per  annum,  J  ;i  '2024709 

Multiply  by  the  principal=  600 


2.  What  is  the  amount  of  01500  for  12  years  at  3^ 
per  cent,  per  annum  ? 

Di-o35=amount  of  Di  for  i  year  at  si  per  cent. 
per  annum. 

And,  i*o3512  x  1500=02266  6oc,  nearly,  Ans. 

ANOTHER  METHOD 

Of  working  compound  intereft  for  years*  months  ,  and  day^ 
which  is  much  more  concife  than  the  preceding  method. 

RULE. 

To  the  logarithm  of  the  principal,  found  in  any  Table 
of  logarithms,  add  the  feveral  logarithms,  anfv.ering  to 
the  number  of  years,  months,  and  days*  found  in  the  follow- 
ing tables,  and  their  fum  will  be  the  logarithm  of  the 
amount  for  the  given  time,  which  being  found  in  any 
table  of  logarithms,  the  natural  number  correfponding 
thereto  will  be  the  anfwer. 


BY  DECIMALS. 


LOGARITHMICK  TAB-LF s,  at  6  per  cent,  per  annum,  for 
Years,  Months,  and  Days. 


Years 

pt-    |  Y. 

•.itc    pts.  |  Month-. 

dec     pts 

i 

025306 

21 

•531426 

I 

•oo  i  1  66 

2 

'0506  1  2 

22 

*5  5^732 

2 

•0043^  i 

3 

•075918 

23 

•582038 

3 

•006466 

4 

•101224 

M 

607344 

4 

•0086 

5 

•12653 

2; 

•63^5 

5 

•010724 

6 

•i  5  1  8  $6 

26 

•657956 

6 

•012837 

7 

•177142 

27 

•683262 

7 

•01494 

8 

•202448 

28 

•708568 

8 

•017033 

9 

•227754 

29 

7^3974 

9 

•0191  jo 

10 

•25306    I  30 

•759^8 

10 

•021  189 

1  1 

•278^66     31 

•784586 

11 

•023252 

12 

•303672 

32 

•809792 

13 

•328978 

33 

•835098 

14 

354284 

34 

860404 

i  <; 

'379  ;9 

35 

•8857. 

16 

•404896 

36 

91  1016 

*  7 

•430202 

37 

936322 

18 

•455508 

38 

•961628 

'9 

•480814 

39 

•986934 

20 

•50612 

40 

I'oi  224 

i);ns. 

1). 

Davs 

i 

•00007  i 

12 

•000857 

22 

•6oiJ58 

2 

•000143 

13 

•000928 

23 

•0010-39 

3 

•©002  i  5 

'4 

•000999 

24 

•oo  1  7  i 

4 

•000287 

1  9 

oo  107 

25 

•00178 

5 

•000,58 

16 

•001  142 

26 

•00185  . 

6 

•000429 

l] 

•oo  i  21  3 

27 

•00192  -< 

7 

•0005 

18 

•001284 

28 

•00199  \ 

8 

•00057  * 

19 

•001355 

29 

•002065 

9 

000642 

20 

•001420 

3° 

•002  3  > 

to 

•0007  13 

21 

•091497 

31               DC        O*1 

W    2 


COMPOUND  INTEREST 

What  is  the  amount  of  £132   i^s.  at  6  per  cent,  per 
annum,  for  9  years,  8  months  and  1 5  days  ? 

To  the  log.  of  /,  132-5=2*  1222 16 
f  Log.  for  9  years  =  -227754 

Add  <  ditto  for  8  months      =  '017033  ' 
(.ditto  for  15  days          =  '00107 


2-368073 

Becaufe  8  months  are  paft,deduft  4!  _.  g 

per  cent,  upon  the  log.  ot  15  days       J  ~ 

Remains     2-3680302 

The  neareft  to  which,  in  the  table  ot  logarithms,  is, 
1-368101,  and  the  natural  number  anfwering  thereto  is 
233-4=^233  8s.  Ans. 

CASE    II. 

When  the  amount  >  rate  and  time  y  are  given^  to  fnd  the  principal* 

RULE. 

Divide  the  amount  by  the  amount  of  £  i  or  D  i  for  the 
given  time,  and  the  quotient  will  be  the  principal. 

Or,  If  you  multiply  the  prefent  value  of  £  i  or  D  i  for 
the  given  number  of  ytars,  at  the  given  rate  per  cent,  by 
the  amount,  the  producl  will  be  the  principal,  or  prcfent 
Forth- 

EXAMPLES. 

i.  What  is  the  prefent  worth  of  757!.  93.  8-£d  due  4 
years  hence,  difcounting  at  the  rate  of  61.  per  cent,  per 
annum  ? 

BY  TABLE  I. 

Divide  by  the  tabula!  ,6,^69)7574861400(^600 

am.  of  il  for  4  years  J 


Bv  TABLE  II. 

Amou.-  =757  486^ 

Mult,  by  the  prefent  wouli  -  t    li.  ]  6 

fo*  4  years,  at  6  per  cent,  per  an.  j 

Ar.s.  599-9999235  S2704-r=rA6oe. 

2    What  principal  n^u;;  be    put   to  intoreft    6  yev.rs,  to 
amount  to  D6^9  42  140^809453  !  25,  at  5^  p^r  cert   per 
? 


BY  DECIMALS. 


CASE  in. 

When  the  principal,  rate  and  amount,  ar?  given,  to  find  the  time. 
RULE. 

Divide  the  amount  by  the  principal  ;  then  divide  this 
quotient  by  the  amount  of  £i  or  l)i  for  i  year,  this  quo- 
tient by  the  lame,  till  nothing  remain,  and  the  number  of 
the  divifions  will  ihow  the  time. 

Or,  Divide  rhe  amount  by  the  principal,  and  the  quo- 
tient will  be  the  amount  of  £i  or  1)1  for  the  given  time, 
which  feck  u  icier  the  given  rate  in  Table  i,  and,  in  a  line 
with  it,  you  will  fee  the  time. 

EXAMPLE. 

In  what  time  will  D$co  amount  to  D6°9  420.  im.-f, 
At  5^  per  cent,  per  annum  ? 


divifions. 


500 

689-42  1  -f 

i  '055 

i  379— 

1-055 

i  -307— 

1-055 

1-239— 

1-055 

1*174+ 

1-055 

1-113— 

1-055 

1-053! 

p  — 

Ans.  6  yearsfc 


CASE   IV. 

When  the  principal,  amount  and  time,  are  given,  to  find  the  ratf 
per  cent. 


Divide  the  amount  by  trie  principal,  and  the  quotient 
will  be  the  amount  of  il.  or  Di  for  the  given  time  ;  then, 
extract  ftich  root  as  the  time  denotes,  and  that  root  will  he 
the  amount  of  il.  or  L).  for  one  year;  t"iom  which  h.b- 
tradt  unitjj,  and  the  remainder  will  be  the  ratio, 

Or,  Having  fouhu  the  amount  of  il  or  D.  for  *:he  time 
2's  above  directed,  io<'k  for  it  in  IVuie  lit,  even  with  the, 


248  DISCOUNT  BY- 

given  time,  and  directly  over  the  amount  you  will  find  the 
ratio. 

EXAMPLE. 

At  what  rate  per  cent,  per  annum  will  0.5:0  amount 
to  1)689  421403+  in  6  years  ? 
689  42  403+ 

__  ---  =1-378843  —  ;  and  /y/6  1-378843  —  =  i  -055. 
500 

Then  i  -055  —  i  ='c$£—  ratio.  Hence  the  rate  is  5^  per 
cent,  per  annum,  Aniwer. 

DISCOUNT  BT  COMPOUND  INTEREST. 

The  f  urn,  or  debt  to    be  dif  counted,    the  time  and  rale^  givent  t# 
Jind  the  prefent  iuoith. 

RULE. 

Divide  the  debt  by  that  power  of  the  amount  of  il.  or 
D  for  year,  denoted  by  the  time,  and  the  quotient  will 
be  the  piefent  worth,  which  fubtracted  from  the  debt, 
will  leave  the  d  fcount. 

EXAMPLES. 

i.  What  is  the  prefent  worth,  and  difcount,  of  600'. 
due  3  years  hence  at  61.  per  cent,  per  annum,  compouad 
intcreft  ? 

_  3 

Divide  by  i-o6|  =  i-i9foi)6oo>o"ooo(503>774i   =  ^03 
J5S-  5^d    prefent  worth,   and  ^000—^503    153. 
^96  4$.  6^d.  =  difcount. 

600  600 

Or,  ----  —  =^503-7741,    and  600 

1-19101  I 

^96-2259. 

By  TABLE   TI. 

In  this  Table,  correA^onding  to  the  time  and  rate,  we  have 
prefent    worth    of    jl.    for  the 


Multiply  by  6oo=debt,  or  principal. 


worth  of  the  debt. 

2.  What  ready  money    will  discharge  a  debt  of 
due  4  years  hence,  at  5  per  cent,  per  annum,  compound 
•  atcreft  ?  Ans.  DB22  700.  zm 


COMPOUND  INTEREST.  249 

ANNUITIES   OR    PENSIONS,   IN  ARREARS, 
AT  COMPOUND  INTEREST. 

CASE    I. 

When  the  annuity,  or  pen/ion,  the  time  it  continues^  and  the  rate 

per  cent,  are  given,  to  find  the  amount. 

RULE. 

1.  Make  i  the  firft  term  of  a  Geo  cetrical  ProgreiJion, 
and  the  amount  of  il.  or  1).  for  i  year  at  the  given  rate 
per  cent,  the  ratio. 

2.  Carry  the  feries  to  fo  many  terms  as  the  number  of 
years,  and  nnd  its  fum. 

3.  Multiply  the  fum  thus  found  by  the  given  annuity, 
and  the  produce  will  be  the  amount  fought. 

Or,  multiply  the  amount  of  £1  or  Di  for  i  year  into  it- 
felf  fo  many  times  as  there  are  years  lefs  by  i  ;  then  mul- 
tiply this  produft  by  the  annuity  ;  and  fubtraft  the  an- 
nuity therefn  m.  Laftly,  divide  the  remainder  by  the 
ratio  lefs  i,  and  the  quotient  will  be  the  amount. 

EXAMPLFS. 

i  .  What  will  an  annuity  of  6ol.  per  annum,  payable 
yearly,  amount  to  in  4  years,  at  61.  per  cent.  ? 

Firjl  Method. 

_  9        _  _  3 

1-4-1*06+  1  *o6|  +  i  o  |  =4-  3  746  1  6-  fum. 
Multiply  by  6c 

362*476960 
20 


953920 

12 


r88i6         Ans.  £262.  95 

9 3 

Or,  i+rc6  4*  i-o6|   -f  i  o6|  X6o=£262  9$.   6£d. 


ANNUITIES,  fcV. 

Second  Method. 
i  •c6xro6xi  -06x1  '06=1-26247 

Multiply  by         60  annuity. 

75-74820 
Subtract  60 

Divide  by  itc6  —  is='o6)i5'7482(26s'47=^>252  ps,  4|d. 

12 


37 

3< 


12 

28 
24 

42 
42 


1*06X1  -06X1  -06X1  -06x60—  60 
Or,  -  -----  =£262-47 

i'o6  —  i 

Or,   by  TABLE    III. 

Multiply  the  tabular  number  under  the  rate  and  op- 
pofite  to  the  time  by  the  annuity,  and  the  prodacT:  will  be 
the  amount. 

2.  What  will  an  annuity  of  6ol.  per  annum  amount  to 
in  20  years,  allowing  61.  per  cent,  compound  intereft  ? 

Under  61.  per  cent,  and  oppofite  20,  in  Table  3,  you 
will  find 

Tabular  number=36*78559 

Multiply  by  6o=annutty. 


2s.  8£d.  Ans. 

3.  What  will  wages,    of.  D  2.5   per  month,  amount  to 
in  a  year;  at  t  per  cent,  per  month  ? 

Ans.  0308  38c,  9m. 


AT  COMPOUND  INTEREST. 


25 1 


CASE   II. 

When  the  amount,  rate  per  cent,   and  time  are  given,  to  find 
the  annuity,  petifion,  &c. 

RULE. 

Multiply  the  whole  amount  by  the  amount  of  £  i  or 
D « ,  for  a  year,  from  which  fubtraft  the  whole  amount, 
divide  the  remainder  by  that  power  of  th;j  amount  of  £  i 
or  Di  fora  year,  fignified  by  the  number  of  years,  made 
lefs  by  unity,  and  the  quotient  will  be  the  anfwer. 

Or,  Find  the  amount  of  an  annuity  oi  £  I  or  D I ,  for  the 
given  time  and  rate  (by  Cafe  I.);  divide  the  given  fum 
by  this  amount ;  and  the  quotient  will  be  the  annuity 
required. 

EXAMPLES. 

i.  What  annuity,  being  forborne  4  years,  will  amount 
10^*262-47696,  at  6  per  cent,  compound  intereft  ? 
1*06 

i -06                         262  47696=amount 
Mult,  by           i  •c6=am.  of  il. 
for  i  yr. 


636 

1060 


157486176 
262476960 

278-2255776 
Subt.  262-47696 

•26247696)  15-7486 176(60!.  Ans- 
157486176 


Subtract 


7146096 
j i 9 i o i 60 

1-26247696 


262-47696x1  -06 — 262-47696 

Or, =60. 

i'o6Xi'o6xi-o6xrc6 — i 


252  ANNUITIES,  fcV. 

Or  thus. 

Amount  of  an  annuity  of    il.  for  4  years,  at  6  per  cent, 

per  aniium,=r4'3746i6  (by  Cafe  I);  and 
262  47696 

—£60  Ans» 

4-374616 

Or,  by  Table  III.  the  amount  of   il.  is  found  to  be 
4-374616  ;   and  the  anfwer  is  found  as  before. 

2.  What    annuity,  being  forborne   20   years,  will   a- 
niount  to  02207-1354,  at  6  per  cent,  compound  intereft? 

Amount  of  an  annuity  of  Di  for  20  years  at  6  per  cent. 
per  antltfiri==3 6*785 59:    And 

Ans. 


CASE  III. 

When  tbs  annuity,  amount  and  rath  are  given^  to  find  the  time. 
RULE. 

Multiply  the  amount  by  the  ratio,  to  this  producl  add 
the  annuity,  and  from  the  fum  fubtrad  the  amount ;  this 
remainder  being  divided  by  the  annuity,  the  guotient  will 
be  that  power  of  the  ratio  fignified  by  thq  time,  which 
being  divided  by  the  amount  of  il.  for  one  year,  and  this 
quotient  by  the  fame,  till  nothing  remain^juhe  number  of 
thofe  divifions  will  be  equal  to  the  time.  Or,  look  for 
this  number  under  the  given  rate  in  Table  i,  and  in  a  line 
with  it,  you  will  fee  the  time.  Or, 

Divide  the  amount  by  the  anuuity  ;  from  the  quotient 
fubtract  i  :  from  the  remainder  fubtracl  the  ratio  ;  from 
fucceffive  remainders,  fubtracl  the  fquare,  cube,  &c.  of 
the  ratio,  till  nothing  remain;  and  the  whole  number  of 
the  fubtra&ions  will  be  the  anfwer.  Or,  find  the  quotient 
in  Table  III.  under  the  rate,  and  in  a  line  with  it  ftands 
the  anfwer. 

EXAMPLES. 

J.  In  what  time  will  6ol.  per  annum,  payable  yearly, 
amount  to  ;£  262^47696,  allowing  6l.  per  cent,  compound 
iilntereft  for  the  forbear an.ce  of  payment ,? 


AT  COMPOUND  INTEREST.  253 

262-47696=3113011111. 
Multiply  by  i-c6=ratio. 

157486176 
262476960 

278-2255776 
Add      60-  =annuity. 


338-2255776 
Subtract        262-47696 

Divide  by  60)75  7486176 

Divide  by  ro6)  1*26247696 

Divide  by  i'o6)i  191016 


Divide  by   i •06)1-1236 
.    Divide  by  ro6)i'c6 

i  Ans.  4  years. 

The  number  of  divifions  by   i  '06,  being  4,  gives  the 
lumber  of  years=4,  the  anfwer. 

-:i         Or  thus  : 
Annuity =60) 262  47 696=amcmnt. 


4-374616 
i.  Subtract    r 


3*374616 

2.  Subtract    i'c6          =ratio. 

2-314616     2 

3.  Subtract    1-1236      =   ratio. j 

1-191016      3 

4.  Subtraft    1-1910.6=   ratio. \ 

Aftiwer  4  years. 

X 


254  ANNUITIES,  ferV. 

Or,  looking  into  Table  III.  under  the  rate,  6,  the  quo- 
tient, 4-37^616,  ftands  againft  4  years,  anfwer,  as  before. 

Or,  in  Table  I.  under  the  given  rate,  you  will  find 
1-262476,  and  in  a  line,  under  years,  you  will  find  4. 

PRESENT  WORTH  OF  ANNUITIES,  ferv.  AT 
COMPOUND  INTEREST. 

CASE    I. 

Whin  the  annuity ,    £5"V.  rate  mid  time  are  gfasv,  to  find  the 

prefent    worth. 

RULE.* 

1.  Divide   the  annuity  by  the  ratio,  or  the  amount  of 
D  i  nr  £  i  for  i  year,  and  the  quotient  will  be  the  prefent 
worth  cf  i  year's  annuity. 

2.  Divide  the  annuity   by  the  fquare  of  the  ratio,  and 
the  quctiert  will  be  the  prefent  worth  for  2  years. 

3.  In  like  manner,  find  the  prefent  worth  of  each  year 
by  itfelf,  and  the  fum  of  all  thefe  will  be  the  prefent  value 
of  the  ar.nuitv,  £/-   L~ 


,  . 

Or,  divide  the  annuity,  &c.  by  that  power  of  the  ratio 
jfignified  by  the  number  of  years,  and  fubtraft  the  quotient 
from  the  annuity  ;  this  remainder  being  divided  by  the 
ratio  lefs  i,  the  quotient  will  be  the  prefent  worth. 

EXAMPLES. 

i  .f  What  ready  money  will  purchafe  an  annuity  of  6ol. 
to  continue  4  years,  at  61.  per  cent.  Compound  intereft  ? 

*  The  amount  of  an  annuity  may  alfo  be  found  for  years  and  parts 
a  of  year,  thus  : 

1.  Find  the  amount  for  the  whole  years,  as  before. 

2.  Find  the  intereft:  of  that  amount  for  the  given  parts  of  a  year. 

3.  Add  this  intereft:  to  the   former  amount,  and  it  will  give    the 
whole  amount  required. 

Tlit  prefent  ivortb  of  an  annuity  for  years  and  parts  of  a  year  may 
be  found  th  us  : 

1    Find  the  prefent  worth  for  the  whole  years,  as  before. 

2.  Find  the  prefent  worth  of  this  prefent  worth,  discounting  for 
the  given  parts  of  a  year,  and  it  will  be  the  whole  prefent  worth  requir- 
ed. 

•j-  Quedions  in  this  cafe  may  alfo  be  anfvvered  by  firft  finding 
the  amount  of  the  given  annuitv  by  Cafe  I.  of  annuities  in  arrears,  p. 
240,  and  then  the  prefent  worth,  or  principal,  by  Cafe  II.  of  Com- 
pound Idereft,  page  245, 


LT  COMPOUND  INTEREST. 


Ans.   207904=^207    185.  c|d. 

S:co::d 


From     60 
Subtract     47*52; 

Bivifor=i'o6  —  1=06)12  475 

207-916=^207   182.  3jd. 

60 

Or,  -=4  =47'525         6o-47-525=:i2'47s 
•p6j 

12-475 

And  --  =207-916. 
•06 

By  TABLE   III. 

Under  61.  per  cent,  and  oppofite  4  we  find 

4  3746[  =  imounc  of  il.  annuiiy  for  4  years. 
Multiply  by       '    6  =iMnuity. 

4766  j=.imount  of  £60  for   4  years. 
Then  oppoiite  4  ye.ir^  and  under  ^'6  psr  cent,  in  Table  2., 

We  have  -792093 
Multiply  by    262  7466 


475*558 

3.68372 

5544651 
1584186 


15.84186 

208'1  197426338=^208    25, 


256  ANNUITIES,  ferV. 

Or,  oppofite  4  years,  and  under  £6  per  cent,  in  Table 
ift,  we  have  i  '26247=^  amount  of  £i  for  4  years 
Then,  262-7466-7-  1-26247:=:  208-  1  209=^208  25.    $d. 

By  TABLE   IV. 

Multiply  the  tabular  number,  under  the  rate,  and  op- 
pofite the  time,  into  the  annuity,  and  the  produft  will  be 
the  prefent  worth. 

Thus,  in  Example  ift.  What  ready  money  -vill  pur- 
chafe  6ol.  annuity,  to  continue  4  years,  at  61.  per  cent. 
compound  intereft  ? 

Under  61.  per  cent,  and  even  with  4  years, 

We  have  3'465i=prefent  worth  of  il.  for  4  years. 
Multiply  by    "       6o=annuity. 


2.  What  is  the  prefent  worth  of  an  annul;  y  of  I)  60 
per  annum,  to  continue  20  yearc,  at  6  per  cent,  compound 
intereft  ?  Ans.  D688-65  (nearly.) 

CASE    II. 

When  the  prefenf  worth,  time  and  rate  are  given,  to  find  the  an- 

Kuity,  rent)  &c. 

RULE. 

1.  From  that  power  of  the  ratio,  denoted  by  the  num- 
ber of  years,  plus  i  ,  fubtract  that  power  of  it  denoted  by 
the  number  of  years. 

2.  Divide  the  remainder  by  that  power  of  the  ratio, 
flgnified  by  the  time  made  lefs  by  unity. 

3  Multiply  the  prefent  worth  into  this  quotient,  and 
the  product  will  be  the  annuity,  penfion,  rent,  &c. 

Or,  i.  Multiply  that  power  of  the  ratio,  denoted  by  the 
number  of  years  plus  i,  by  the  prefent  worth. 

2.  Multiply   that  power  of  the   r-itio,   denoted   by  the 
time,  by  the  prefent  worth,  and  i'ubtracT;  this  product  from 
the  former. 

3.  Divide  the  remainder  by  that  power  of  the  ratio,  de- 
noted by  the  time  made  lefs  by  unity,  and  the    quotient 
will  be  the  annuity. 

EXAMPLES. 

i.  What  annuity,  to  continue  4  years,  will  £207*904. 
purchafe,  compound  intereft,  61.  per  cent,  ? 


AT  COMPOUND  INTEREST. 

Firft  Method. 

Fromro6xi-o6Xr-o6Xi-o6Xrc6=i  3382255776 
Subt.  *  06xro6Xro6Xro6          =i  ^6247696 

4 

Divide  by  rc6l   —  i-' 


•2885898 
Multiply  by          207  9  prefent  worth. 

25973082 
20201286 
57717960 


Ans.  59-9978:942=^60 

Second  Method. 

From i  06x1  06X1 -06x1-06x1 '06X2 ^7-9=278-2 17097573 
Take  ro6xi  06X1*06X1-06X207  9          =262  2689590,^4 


Divide  by   ifo6|  — 1=  2624769^)1  y  748« 

Qjotient=59  998=^60. 
By  TABLL   V. 

Multiply  the  tabular   number,  correfponcling   with  the 
rate  and  tim.,  by  the  purchife  money,   and  the  produfl 
will  be  the  annuity 
Under  61    per  cent  md  oppofite  4  years,  you  will  find 

•2S^59=innuity  which    il.    will  purchafe  in 
Multiply  by   207-9  "  L.4  yea-rs« 

259731 

20i013 

5771*0 

59997861  =  ^60   Ans. 

2.    What  (alary  to  continue  20  years,  will  D688  95*2. 
purch.iie,  at   6  per  cent,  compound  mtereft  ?     Ans.  D63. 

CAS'L  III. 

Whsn  the  avtxiy,  prcfint  vji'-th   and  ratht  are  given,  to  Jtid 

the  tinis. 

RULE. 

Divide  the  annuity  by  the  pro Ju^.  of  the  prefe.it   worth 
and  rat:')  «ub;racl?d  iro*n  the  fum  of  the  preient  vrorth  and 

X    2 


ANNUITIES,  CsV. 

annuity,  and  the  quotient  will  be  that  power  of  the  ratify 
denoted  by  the  number  of  years,  wh-ch,  being  divided  by 
the  ratio,  and  this  quotient  by  the  fume,  till  nothing  re- 
rn;:in,  the  number  of  divifions  will  fhow  the  time  :  Or, 
the  above  quotient  being;  fought  in  Table  i!t  under  the 
given  rate,  in  aline  with  it,  you  will  fee  the  time. 

EXAMPLES. 

i.  For  how  long  miy  an  annuity  of  6ol.  per  annum  be 
purchufed  for  £207 -906336762,  at  6  per  cent,  compound 
intereft  ? 

Multiply  207-906336762 
by  i  *o6 


12474380^72 
2079063307620 


220  38071696772 

To  207*9063 36762=prefent  worth 
Add     60 '  =  annuity. 


From       267  906336762 
Take       220  '$80716967 

47  •  5  2  ^  6  1  9-79  5=divifor. 
47*525  6  1  9795)60*000000000(1  '26  247696 
Divide  by  i'c6)i  26247696 


i'o6)  i  -1236 

___ 

1*06)  i  -06 

The  number  of  divifions= 
^     time=4  years. 
60 

Or,  -  -  .....  -—  •    --------  = 

207  9063367624-60  —  207-906536762  x  1*06 

1*26247696,  which  being  fought  m  Table  I,  under  the 
given  i  ate,  tn  a  line  with  it,  is  4=4  years. 


AT  COMPOUND  INTEREST.  259 

2.  How  long  may  a  leafe  of  0300  yearly  rent,  be  had 
for  Dzi3r$4  ,  allowing  5  per  cent,  compound  intereit, 
to  the  purchafer  ?  Ans.  9  years. 


ANNUITIES,   LEASES,  &c.    TAKEN    IN  RE- 
FERS1&N  AT.  COMPOUND  INTEREST. 

When  the  annuity  ^  time  and  rafic,  are  given,  to  find  the  prefer.* 
worth  of  the  annuity  in  r ever/ion. 

RULE. 

1.  Divide  the  annuity  by  that  power  of  the  ratio  denot- 
ed by  the  time  cf  its  continuance. 

2.  Subtract  rhis   quotient  from  the  annuity  :  divide  by 
the   ratio  lefs  i,  and  the  quotient  will  the  prefent  worth,, 
to  commence  immediately 

3.  Divide  this  quotient  by   that  power  of  the  ratio  de- 
noted by  the  time  of  reveriion,  (or,- time  to  come,  before 
the  annuity  commences)  and  the  quotient  will  be  the  pre- 
fent worth  of  the  annuity  in  reverfion. 

Or,  i.  Multiply  the  annuity  by  that  power  of  the  ratio 
denoted  by  the  time  of  its  continuance,  minus  unity,  tor 
a  dividend. 

2.  Multiply  that  power  of  the  ratio  denoted  by  the 
time  of  its  continuance,  that  power  of  it  denoted  by  the 
time  of  feverfion,  and  the  ratio  lefs  i,  continually  rogeth- 
er  for  a  divifor,  and  the  quotient  arifiug  from  the  divi- 
fioii  of  thefe  two  numbers  will  be  the  preient  worth  of  the 
annuity  in  reverfion. 

EXAMPLES. 

i  What  is  the  prefent  worth  of  6ol.  payable  yearlyy 
for  4  years ;  but  not  to  commence  till  two  years>  hence* 
at  61.  per  cent.  ? 

Firrt  Method. 

Ratio=ifc6 

ro6 


2d  power=  1-1236  Carried 


2*>»  ANNUITIES,  &V. 

3d  power—i*  i  2*6 

11!!! 

67416 
55708 
2'472 
11236 
1 1 2  3  6 

Divide  by  4th   power— i  26247696)60  cooooooooooo( 
Subtraft  the  otuotient=47'5256i979428i 

Divide  by  i  06 — 1—06)12-474380205719 
Divide   by    1*06X1  -06=1-1236)207  9c6336:6ig( 

Quotients:  185  035899= 
£185  os.  S^d.^prefent  worth  oi  the  annuity  in  rcvertion. 

Or,  in  Table  IV.  find  the  prefent  value  of  il.  at  the 
given  rate,  both  for  the  time  in  being  and  the  time  in  re- 
veriion  added  together,  and  fubtraS  the  prefent  worth 
of  the  time  in  being  from  the  other,  multiply  the  re- 
mainder by  the  annuity,  and  the  produd  will  be  the 
anfvver. 

Prefeut  worth  of  the  time  in  b^in^  and  reverfion=4'9 17^2 
Prefent  worth  of  the  time  in  being=i  ^333 

3-8402 
60 

Ans.  ^185  04120 


60  60  —  47*5296 

Or,  ----  =47-5256         ---  ~  -  =207-906 


207*906 

And  ----  =  185*035899 
1-1236 


AT  COMPOUND  INTEREST.  261 

Second  Method. 

•26247696=41?!  power  —  i 
Multiply  by  6;~anr.uity. 

15-74861  7  6o=dividend. 

i  -26  247696=4^  power, 
i'i2:;6-zd  power, 

757486176 
378743088 


126247696 
126247696 

I  '41  85  19  I  I  22  ,'6 

•c6:=  ratio  —  I 

•085  1  1  1  1467  ?  c  36=rdhrifor. 

•0851  ii  15)  15  -7486  17  60(185-036  Ans, 


Or,  ---  -  -------  -=185-036 

"FZ6|4  ^T^l*  x  1-06—1 

2.  What  is  the  prefent  worth  of  a  reverfion  of  a  leafe 
of  D6o  per  annum,  to  continue  20  years  hut  not  to- 
commence  till  the  end  of  8  years,  allowing  6  per  cent  to 
the  purchafer  ?  Ans.  1^43  1-782  (nearly.) 


An  annuity,  fiver al  times  in  re-verjlbn^  and  rate  being  given,  to 
find  the  fevsral  prefent  values. 

F,:nd  the  prefent  value  of  £i  or  Di  by  Table  4,  at  the 
given  rate,  and  for  the  feveral  given  times,  which,  being 
feveraily  multiplied,  by  the  annuity,  the  products  will  be 
the  feveral  prefent  values  of  that  annuity,  for  the  feveral 
times  given  :  fubtract  die  feveral  prefent  va  ues,  the  one 
from  the  other,  and  the  feveral  remainders  will  anfwer  the 
queition. 


2<5z  ANNUITIES,  feV.. 

3.  A  has  a  term  of  6  years  in  an  eflate  at  6ol,  per 
annum.  B  has  a  term  of  14  years  in  the  fame  eftate,  in 
reverfion,  after  the  6  years  are  expired  «•  and  Chis  a  fur- 
ther term  of  1 6  years,  after  the  expiration  of  20  years.  I 
demand  the  prefent  values  of  the  feveral  terms,  at  6  per 
cent.  ? 

£     s.    d. 

Prcf.  value  of  ^'t  for  36  yrs=  14-6 172  2x60=87  7  o     7^- 
Ditto  of  ditto  for  20  years  =i  i '46992x60=688  3    io£ 
Ditto  of  ditto  for  6  years    =  4-91732X60=295  o     9;{-  = 
AJs  term.       Therefore,    877  o  7^—688  3    io|=^i88 
163.9,1  C's  term,  and    688  3   io| — ^95  o  9^=^393  3 
ii=B's  term. 

4.  For  a  leafe  of  certain  profits  for  7  years,  A  offers  to 
pay  D3oo  gratuity,  and  0500  per  annum,  B  c/fFcrs  D8oo 
gratuity  and  D25O  per  annum,  C  bids  D  1:300  gratuity 
and  D2OO  per  annum,  and  D  bids  Di^oo  for^the  wiiole 
pur  chafe,  without  any  yearly  rent  ;  which  is  the  beii  offer, 
computing  at  6  per  cent.  ?  D. 

By  Table  IV.  the  prefect  worth  of  0300  per  1    l6 

annum,  for  7  years,  at  6  per  ce;  t.    L 

To  wi  icii  ,idd          300* 

Value  oi  A's  offe.-  = 


Prefent  worth  of  Di$c  per  annum  for  7  years=i395  595 

To  which  add     boo- 

Value  of  B 'softer- 2 1 95  595 

Prefent  worth  cf  Duos  per  annum  for  7  years=i    16-476 

To  which  add     1300' 

Value  of  C's  offer —2  4 16-47 6 

D's  offer 
Hence  it  appears  that  D^s  offer  is  the  bed. 


The  above  queftions  may  be  anfwered  by  the  4th  and 
ad  Tables. 


AT  COMPOUND  INTEREST.  263 

Take  quefticn  ift  for  example. 

T.  Multiply  the  tabular  number  in  Table  4,  corre- 
fpondin^  to  the  rate  and  the  time  of  continuance,  into 
the  annuity,  and  the  product  will  be  the  prefent  worth, 
to  commence  immediately. 

2.  Multiply  this  prefent  worth  by  the  tabular  number 
in  Tr.-ble  2,  correfponding  to  the  rate  and  the  time  of  re- 
verfion,  and  the  produci  will  be  the  prefent  worth  of  the 
annuity  in  reveifion. 

In  Table  4th  we  have  3*465 1 

Multiply  by  6.  ^annuity. 

207-9060 
In  Table  2  we  have  '889996 

1247436 
1871 154 
1871154 
1871154 
1663248 
1663^48 


worth   of  there- 
verfion. 


264 


TABLES. 


TABLE  FIRST; 

Showing  the  amount  of- il.  or  D.  i  from  i  year  to  forty 
years. 


ys.jq'  per  cent  (6  per  cent.]  ys  (5  per  cent. 

6  per  cent. 

2 

3 
4 

5 

"i  050000 
I-I02500'' 
I  1576250 
I'2I5506 
1*276281  : 

rooooooo 
i  'i  236000 
11910160 
['2624769 
i  3382256 

2112-7859625 

22  2'9  25  260; 

2413-2250999 
25  3'3863549 

3'3995636 
360353741 
3  8197496 
4-0489346 
4-2918707 

6 

7 
8 
9 
10 

•3400956 
•4071004 

'4774554 
5513282 
•6288946 

i  4185  191 

1*5036302 
1*5938480 
16894789 
1-7908476 

26 

27 
28 
29 

3° 

3  5556726 
3'7334563 

4-1161356 
4  3219423 

4-7493829 
4-8223459 
5-11  16867 
5-4183879 
5*74349'  2 

1  1 

12 

'4 
15 

'7I03393 
;  -7958563 
•8856491 
9799316 
2-0789281 

i  8982985 
2  012196  ; 
2-1329282 

2  2609039 
2-3965581 

£  5403516 

'-6927727 

3  ^25599.- 
)3'2o7i355 

31 
35 

4-5380394 
4  7649414 
5  0031885 

5  2533479 
5-5160152 

6-0881007 

6-4533867 
6  8405899 

7-2510.753 
7  '':86o868 

16 

17 
18 

'9 

2C 

2-18287  * 
i  29201  8 

2*5269502 
'26532977 

37 

39 

,40 

5  7918101 
6  0814069 
6-385*772 

-.7047511 

8  147252 
8  6860871 
9-1542523 
9-7035074 
10-2857178 

TABLES. 


265 


TABLE  SECOND. 

Showing  the  prefent  value  of  il.  or  D.I  due  at  the  end 
of  any  number  of  years,  from  i  to  40. 


years.  15  per  ct  |6  per  ct 

years.  (5  per  ct.|6  per  ct. 

l 
2 

3 
4 
5 

•952381 
•90703 
•863838 
•822702 
•783526 

94S396 
•889996 

•889619 
•792093 

•74/258 

21 

22 

23 
24 

25 

358942 
34185 
•325571 
•3  10068 

3°53°3 

•294155 

•277505 
•261797 
•246978 
232998 

6 

7 
8 

9 

10 

•746215 
•710681 
•676839 
644609 
•613913 

•584679 

•556837 
•530321 
•505068 
•481017 

70496 
•665057 
627412 
•594898 
558394 

26 

27 

28 

29 

3° 

281241 

•267848 
255094 
•242946 

23137? 

-220359 
209866 
199872 

J9°355 
•18129 

21981 
•207368 
•<9563 

•184556 

1741  1 

1  1 

12 

'3 

14 
15 

•562787 
496969 
•468839 
442301 
•417265 

31 

3* 
33 
34 
35 

•164  55 

1  5495  7 
146.86 
137912 
130105 

16 

'7 
18 

'9 

20 

•458311 
•436297 
•4I552I 

'395734 
•376880 

•393647 
•371364 

•35°343 
'33°5'3 
31  1804 

3^ 
37 
38 

39 

40 

•172057 
•164436 
•156605 
•1491^8 

142046 

•122741 

11  5793 
109182 
•103002 

•097.72 

266 


TABLES, 


TABLE  THIRD. 

Showing   the  amount   of    il.    or  D.I  annuity   for  any 
number  of  years  from  I  to  40. 


ys.|5  per  cei)t.]6  per  cent.|jysj 

5  per  cent.  |  6  per  cent. 

i 

2 

3 
4 
5 

6 

7 

9 

10 

i 
205 

3  -'5*5 
4310/25 

5-525631 

l1 

2-06 
3^M 
4-474616 
5637093 

21 

22 

23 
24 

>5 

35  7'9*52 

38  505214 

4<l'.43°47,5 

44-501999 

47-727009 

39-992725 
43-392289 

46*995826 
50-8.5576 
54^6451 

6   Soi-y  13 
8M42OO8 

9  -549109 
1  1-026564 
12577892 

6-97^318  26 

8-398387  27 
9-897467   28 

"•49l3l5\i9 

13-180794)30 

51-113454 
54  669126 
58-402583 

62-3227  12 
60-438847 

59-156381 

63*7°5763 
68-528  109 

73  <S39796. 
79-058183 

1  1 

12 

J3 
J4 

15 

14-206787 
15*917  126 
17-712983 
19598632 
21-578563 

14-971642 
1  6  86994 
18-882132 

21  01506  j. 

23  275968 

31 

8 

34 
35 

70  76079 

75  298829 
80  063771 
85-066959 

90-320307 

84-801674 
90889775 
97-343161 

[04-183751 
111-434776 

i  19-120803 
i  27-2681  14 
135-904201 

*45'°58453 
154-761961 

16 

•7 
18 

19 

20 

23657492 
25  840366 
.8-132385 
30-529004 
33  oft  59s  4 

25-67252: 
28-212879 
3-0905652 

W759991 

36-78;^- 

36 
37 
38 

39 

(40 

95  836323 

101-6281  ^<; 
107-709546 

I  i  4-09  1-025 
!  20-799774 

TABLES. 


TABLE  FOURTH. 

Showing  the  prefent  worth  of  il.  or  D.I  annuity,  for  any 
number  of  years  from  i  to  40. 


Hi 

5  ptr  ce   t|6pv.r  cent[[  yrs.  {5  percent 

6  per  cent. 

i 

o-9;23?5i  0-94339 

:    21 

12-82115    11-76407 

2 

1-85911 

I<83339 

!    22 

13-163        12-01158 

3 

272325 

2-67301 

23 

13-48807    12-30338 

4 

3*5  459>' 

V4^>« 

24 

13-79864;   12-55035 

5 

4-32918 

4-21236 

2  5 

,409394    12-7833,- 

6 

5-07569 

4-9673-2 

26 

14-^7518'   13-00316 

7 

5-78037 

5-58238 

27 

14-64303    13-21053 

8 

6-46321 

6-70979 

28 

14-8^^13    1  3  -406  '6 

9 

7-10782 

6-80  '69 

29 

15-14107 

13-59072 

10 

772173    7-36-08 

3° 

I5'37245 

13-76483 

1  1 

8-3  O-HJ 

7-88687 

3l 

15-59281 

13-92908 

12 

8-8632- 

8-38384 

32 

15-802.68 

14  08398 

13 

M 

9*39357 
9-89864 

885268    33 
9-29498    31 

16  00255    14-22917 
16-1929      14-36613 

15 

10-37966 

9-71225 

35 

16374.9 

•4495^3 

16 

[0-83777 

)O'iO589 

36 

16-54685     i4'6i722 

ll 

ii  27407 

(0-47726 

3" 

16-7  1  129!   J  4'7321 

18 

11-68958 

10-8276 

38 

16-86789    14-84048 

'9 

12  0853; 

1  1-1581  ii 

39 

17-01704    14-9427 

20 

12  4622 

•  i  -4.^992    40 

(7  1^909    i  <;-o}9i3 

268 


TABLES. 


TABLE  FIFTH. 

""he  annuity  which  il.  or  D.i  will  purchafe  for  any  num- 
ber of  years  to  come,  from  i  to  40. 


(fl*S« 

5  per  cent 

6  per  ct  |[  vrs.  |  5  per  ct. 

6  per  ct. 

i   ii-o^ 

106 

2  I 

•078 

•085 

2 

'5378 

'54544 

22 

•07597 

•08303 

3 

•36721 

'374^ 

23 

•07414 

•08128 

4 

•28201 

•28859 

24 

•07247 

07968 

5 

•23097 

•23739 

25 

07095 

•07823 

6 

-19702 

•20336 

26 

06956 

•0769 

7 

•17282 

•*79*3 

27 

•06829 

•°757 

8 

•IS473 

•16103 

28 

'067  I  2 

*°7459 

9 

•14069 

•14702 

29 

'06604 

•07358 

10 

•i  295 

•'3587 

3° 

•06505 

•07272 

1  1 

•12039 

•12679 

31 

•06413 

•07179 

12 

•i  1282 

•11927 

32 

•06328 

•071 

*3 

•10645 

•i  1296 

33 

•06249 

•07027 

14 

'IOIO2 

-10758 

34 

•06175 

•06959 

15 

'09624 

•10296 

35 

•06107 

•06899 

16 

•09227 

•09895 

3* 

•06043 

•06839 

I? 

•0887       j    '09544 

37 

.05984 

•06785 

18 

•03555 

•09235 

33 

•05928 

•°<$735 

19 

•082/4 

•08962 

39 

•05876 

•06689 

20 

•08024 

•08718 

40 

'05828 

•06646 

ALUG  \TION.  2i5j» 


ALLIGATION 

IS  the  method  of  mixing  two  or  more  fimples  of  differ*, 
ent  qualities,  fo  that  the  compofiti'  n  may  be  of  ,t  me.m  or 
middle  quality  :  It  confifts  of  two  kinds,  viz.  Alligation. 
Medial,  and  Alligation  Alternate. 


ALLIGATION  MEDIAL 

Is,  when  the  quantities  and  prices  of  feveral  things  are. 
i,  to  find  the  mean  price  of  the  mixture  compounded, 
of  thofe  things. 

R.ULE. 

As  the  fum  of  the  quantities,  or  the  whole  compofition, 
is  to  their  total  value  ;  fo  is  any  part  of  r,he  compofition 
to  its  mean  price  or  value. 

EXAMPLES; 

i.  A  Tobacconift  would  mix  6olb.  of  tobacco  at  6J, 
per  Ib.  with  5olb.  at  is  4olb,  at  is.  6d  and  3olb.  :it  zs> 
perlb.  :  What  is  lib.  of  this  mixture  worth  ? 

Ib.       s.  d.     £  s.  Ib,       £       Ib. 

60  at  o  6  is   i    10     AJ  180  :    ic  ::   i 
50  —  i   o —  2    10  i 

40  —  i   6  —  30  — 

3  c  —  20  —  30  10 

9um  of  the!  —  20 

fimples      j  1 80      Total  10     o. 

180)200(13, 
1 80 

20 
12 

I  80) 2 40(1  id. 
180 

4.ns.  is.  ij-d,  

60 


ALLIGATION. 

2.  A  farmer  would  mix  20  buihelsof  wheat  at  Di  per 
builiel,  1  6  bufhels  of  rye  at  750.   per  i.-utlisl,    12  buihels  of 
barley  at  jjcc.  per  bu{hel,and  8  bulliels  of  oats  at  4<Dc.  per 
buflicl  :  What  is  the  value  of  one  buihel  of  tnis  mixture  ? 

Ans.  730    5-'5-m» 

3.  A  wine  merchant  mixes    12  gallons  of  wine  at  7">c. 
per  gallon,  with  24  gallons,  at  900.  and  16  gallons  at  Df 
loc.   :   What  is  a  gallon  of  this  composition  worth  ? 

Ans.    920.  6rn. 

4.  A    goldfmith  melted  together  8oz.  of  gold   of  2  2 
carats  tine,   ilb.  8oz.  of  21  carets  fine,  and    icoz.  of    18 
carats  fine  :    Pray  what  is  the  quality,  or  rmenefs  of   the 
competition  i 

8X224-20X21  -f  icXi8 
Ans.   —  -  -------  =2  ofV  carats  fine. 


$.  A  refiner  melts  5lb.  of  gold  of  20  carats  fine  with 
Sib.  of  1  8  carats  fine  :   Ho.v  much  alloy  muit  be  put  to  it 

ID  rnake  it  22  carats  line  ? 


22— 5X20  4-tix'i8--:-5+8= 
Anfwer.     It  is  not   rine  enough   by  3^3-  carats,  fo  that 
no  alloy  muit  be  added,  but  more  goid. 


ALLIGATION  .ALTERNATE* 

IS  the  method  of  finding  what  quantity  of  each  of  the 
ingredients,  whofe  rates  are  given,  will  compofe  a  mixture 
of  a  given  rate  :  fo  that  it-is  the  reverie  of  Alligation  Me- 
dial, and  may  be  proved  by  it. 


*  By  connecl!n<r  the  !efs  rate  with  the  greater,  and  placing  the 
difference  between  them  and  the  mean  rate  alternately,  or  one  after 
'he  other  in  turn, the  quantities  refulting  are  fuch,  that  there  is  pre- 
,:ifely  as  much  gained  by  one  quantity  as  is  lott  hy  the  other,  and 
Therefore  the  gain  and  lofs  upon  the  whole  are  equal,  and  are  exadlly 
tlu  propofed  rate. 

'    In  !i  :e  ma  -tier,  let  the  number  of  finaptes  be  what  it  may,  and  with 
-    -  -T,  irjT-  fr?"v?r.  e?.ch  one  is  linked, iince  it  is  always   &  lefs  with  a 


ALLIGATION.  271 

CASE  I. 

The  whole  work  of  this  cafe  confitls  in  linking  the  ex- 
tremes truly  togei:h?r  and  tikin^  the  differences  between 
them  and  the  mean  price,  which  differences  are  the  quanti- 
ties fought. 

RULE. 

1.  Place  the  feveral  prices  of  the  fimples,  being  reduced 
to  one  denomination,  in  a  column  under  each  other,  the 
le;n<-  upper  m  ft,  and  fo  gradually  downward,   as  they  in- 
creafe,*  with  .t  line  of  connexion  at  the  ieft  hand,  and  the 
mean  price  af  the  left  han-.i  of  *li. 

2.  Connect,  with  a  continued  line,  the  price  of  each  fim- 
ple,  or  intirt  di^nt,  which  is  lefs  thai)  that  of 'he  compound, 
wuh  one  or  any  i: umber  of  thofe,   which  are  great*. r  than 
the  compound*  =md  rach  greater  rate  or  price  with  one  or 
any  number  of  h-  lefs. 

3.  Place  the  difference,   between  the    mean    price,   or 
mixture  rate,  and  that  of  each  of  the  fimples,   oppoihe  to 
the  rates  with  which  they  are  connected 

4.  Then,  if  only  one  difference  (land  ag^inH  any  rate,  it 
will  be  the  quantity  belonging  to  that  rate  ;  but  if  there  be 
fevera),  their  ium  will  be  the  quantity. 

EXAMPLE?. 

i.  A  merchant  has  fpice>,  fome  at  is.  6d.  per  Ib.  fome 
at  2S.  fome  at  45.  and  fome  at  55.  per  Ib  :  How  much  of 
each  fort  mult  he  mix  that  he  m-,y  fell  the  mixture  at  35. 
4d.  per  Ib.  •? 


greater  than  the  mean  price,  there  will  he  an  equal  balance  of  lofs 
and  gain  between  every  two,  and  consequently  an  equal  balance  on 
the  whole. 

It  is  obvious  from  the  rule,  that  queftions  of  this  fort  admit  of  a 
great  variety  of  anfwers  •,  for  having  found  one  anfwer,  we  may  find 
as  m.uw  more  as  we  i)leai"e,  by  only  multiplying  or  dividing  each  of 
the  quantities  found,  by  2,  3,  4,  &c.  the  reafon  of  which  is  evident  ; 
for  ;f  two  quantities  of  two  fimples  make  a  balr.ice  of  lofs  and  gain, 
\vithrefpect  to  the  mean  price,  Ib  muft  alfo  the  double  or  triple,  the 
half  or  third  part,  or  any  other  ratio  of  thei'e  quantities,  and  fo  on  ad 
infiiiitmn. 

If  any  one  of  the  fimples  be  of  little  or  no  value  with  refpect  to  the 
reft,  its  rate  is  fuppofed  to  be  nothing,  as  v/ater  tiixsd  with  wine,  aod 
alloy  with  gold  ajid  Hlver. 


L?  2 


Mean  rate 


ALLIGATION. 


8  at   i    6") 
28  —  2  o  I         ., 

38^4ofPer1^ 
16—  5  oj 


d. 


NOTE.     Thefe  feven  anfwers  arife   from  as  many 
\:i©us  ways  of  linking  the  fimples  together* 


ALLIGATION.  273 

2.  *A  merchant  has   Canary  wine,   at  35.  per   gallon, 
Sherry  at  2s    id.  and  Claret  at  is.  $d.  per  gallon  :   How 
much  of  each  fort  muft  he   take  tu  fell  it    at  2s.  4d   per 
gallon  ? 

f  36^    3+*i        14  at  35.  od  1 
Mean  rate  28d.  4  2j  JV  8  8  —  21      !•  per  gal. 

l^J  8  a_!    s   J 

3.  Agoldfmith  would  mix  gold  of  19  carats  fine,  with 
forae  of  1 6,  18,  23,  and  24  carats  fine  fo  that  the  compound 
may  be  2  i  carats  fine  :  What  quantitity  of  each  mult  he 
take  ? 

Ans.  5oz.  of  1 6  carats  fine.  507.  of  18,  5oz,  of  19,  10 
oz.  of  23,  and  ID  oz  of  24.  carats  fine. 

4.  It  is  required  to  mix  feveral  forts  of  wine,  at6oc.  900 
and    D.i   i5c.    per  gallon,    with  water,   rluit  the  mixture 
may  be  worth  75  c.  per  gallon  :  Of  how  much  of  each  fort 
muft  the  competition  cor. fill  ? 

Ans- 40  gallons  of  water,  15  galls,  of  wine  at  6oc.  15. 
galls,  do.  at  9oc.  and  7^  galls,  do.  at  D.  i  15. 

CASE    II. 

When  the  rates  of  all  the  ingredients,  the  quantity  of  but  one 
ofthsni)  and  the  mean  rate  of  the  whole  mixture  are  given* 
to  find  the  feveral  quantities  of  the  reft,  in  proportion  to  the 
quantity  given. 

RULE. 

Take  the  differences  between  each  price,  and  the  mean 
rale,  and  place  them  alternately,  as  in  Cafe  I.  Then,  as 
the  difference  ftanding  again  ft  that  fimple,  whofe  quanti- 
ty is  given,  is  to  that  quantity,  fo  is  each  of  the  other  dif- 
ferences, fcverally,  to  the  ftveral  quantities  required. 

EXAMPLES. 

i.  A  merchant  has  4oib.  of  tea,  at  6  (Hillings  per  Ib. 
which  he  would  mix  with  fome  at  55  8d.  fome  at  55.  zd. 
and  fome  at  45.  6d  :  How  much  of  each  fort  muft  he  take 
to  mix  with  the  4010.  th.i'c  he  may  fell  the  mixture  at  53. 
5d.  perlb. 

*  Note,  the  fecond  queftion  admits  but  of  oue  way  of  linking,  and 
fo  but  of  one  anlwer  ;  yet  all  numbers  in  the  fame  proportion  between 
themfelves,  as  the  numbers  which  compofe  the  anfwer,  will  likewife 
fatisf  the  condition  of  the 


,74  ALLIGATION. 


ftands  againft  the  given  quantity. 
fio  :   28T8^  at  45.  6H.> 
:<  10  :  28^-—  5      2     5- per  Ib. 
(.14  :  40      —  5     »    J 
2.  How  much  gold  of  16,  20,  and  24  carats   fine,   and 
how  much  alloy,  imift  be  mixed  with  ?o  oz,  of   18  carats 
fine,  that  the  composition  may  be  22  carats  fine. 

Ans.  looz.  of  1 6  carats  fine,  10  of  20,  170  of  24,  and 
10  of  alloy. 

ALTERNATION  TOTAL. 

CASE  III. 

When  the  rates  of  the  feveral  ingredients,  the  quantity  to  Is 
compounded,  and  the  mean  rate  of  the  while  mixture  are 
given,  to  find  how  much  of  each  fort  will  make  up  the  quan- 
tity. 

RULE. 

Place  the  differences  between  the  mean  rate,  and  the 
feverai  prices  alternately,  as  in  Cafe  I.  ;  then,  as  the  fum 
of  the  quantities,  or  differences,  thus  determined,  is  to  the 
given  quantity,  or  whole  compolition  ;  fo  is  the  difference 
of  each  rate,  to  the  required  quantity  of  each  rate. 

EXAMPLES. 

i.  Suppoffc  I  have  four  forts  of  currants,  at  8d.  I2d. 
i8d.  and  zzd.  per  Ib.  ;  the  word  will  net  fell,  and  the  beft 
are  too  dear  ;  I  therefore  conclude  to  mix  i2olb.  and  fo 
much  of  each  fort  as  to  fell  them  at  i6d.  per  ib.  :.  How 
much  of  each  fort  muft  I  take  ? 

d.         Ib.  Ib.    Ib.       d. 


i6d. 


Sum=2o  1  20 


3.   A  goldfmith  has  feveral  forts  of  gold;  viz.    of   16, 
17,  20  and  22  carets  fine,  and  would  melt  together,  of  all 


ALLIGATION.  275 

thefc  forts,  io  much  as  may  make  a  mafs  of  4OOz.  12  ca- 
rats fine  ;  how  much  of  each  fort  is  required  ? 

Ans.  i6oz  15  carats  fine,  8oz  17,  4oz,  20,  and  i2oz. 
of  22  carats  firre. 

3.  A  merchant  would  mix  four  forts  of  wine,  of  feveral 
prices,  viz.  at  75c.  D  1  2$c.  D.i  coc.  and  D  i  62-gC  per 
gallon  ;  of  thefe  he  would  have  a  mixture  of  72  gallons, 
worth  D.i  37gC  per  gallon  ;  what  quantity  of  each  fort 
murt.  heh.-'vr. 

Ans  8  at  75c.  16  at  Di  25c.  40  at  D.i  ^oc.  and  8  at 
D.i  62  'c.  Or,  1  6  at  75c.  8  at  D.i  ijc.  8  at  D.i  500. 
and  40  at  D.  i 


CASE     IV. 
When  wore  than  one  of  the  Simples  are  limited. 

RULE. 

Find,  by  Alligation  Medial,  what  will  be  the  rate  of  a 
mixture  made  oi  the  given  quantities  of  the  Twitted  Jimples  ori- 
ly  ,  then,  confider  this  as  the  rale  of  a  limited  fin:  pis  ,  whofe 
quantity^  the  fum  of  the  fir  it  given  limited  fimpis;  .  from 
which,  and  the  rates  of  the  unlimited  Jimples,  by  Cafe  2d« 
calculate  the  quantity. 

EXAMPLES. 

i.  How  much  wine  at  boc.  and  at  874-0.  per  gallon, 
muft  be  mixed  with  8  gallons  at  75c.  and  12  gallons  at 
pec  per  gallon,  that  the  mixture  may  be  worth  Sz^c.  per 
gallon  ? 

T  •    •    j  r      1        T    8  gallons,  at  7cc.  =D.   6 
Lmuud  fimples  8 


20  l6    80 

Gal.     D.  c.     Gal.     c. 
As  20  :    1  6  80  ::    i   :    84  per  gallon. 
Now,  having  fou-.id  the  rate  of  the  limited  fimples,  the 
cmcftion  may  (tand  thus  :    How  much   wine,  at  8oc.  and 
8?ic-  P^r  gallon  muft  be  mixed  with   20  gallons  ?.t  84C. 
per   gallon,   that-the    mixture   may    be  worth  82^  per 
gallon  ? 


8o 


^-  gallons  at  8oc. 

l 


276  POSITION. 


Proof. 

52     gallons  at     8cc,     =  D-4i  6cc. 

5o     -  ----      87^     =  17  50 

8     _  ---      75       =  6 

12     ----     90      •=  10  80 

92     --  —  —     82  J    =  75  oo 


POSITION. 

POSITION  is  a  rule,  which,  by  falfe  or  fuppofed  num- 
bers,  taken  at  pleafure,  difcovers  the  true  ones  required. 
It  is  divided  into  two  parts  •,  Jingle  and  double. 

SINGLE  POSITION. 

Single  Pofition  teaches  to  refolve  thofe  queftions,'  whofe 
refults  are  proportional  to  their  fuppofitions  ;  fuch  are 
thofe  which  require  the  multiplication  or  divhion  of  the 
number  fought  by  any  propoied  number  ;  or  when  it  is 
to  be  increafed  or  diminifhed  by  itfeif  a  certain  propofed 
number  of  times. 

RULE. 

1.  Take  any  number,  and  perform  the  fame  operations 
with  it  as  are  dcfcribed  to  be  performed  in  the  queflion. 

2.  Then  fay,  as  the  fum  of  the    errours  is  to  the  given 
fum,fo  is  the  fuppofed   numb.-  to  the  true  one  required. 

Proof.  Add  the  feveral  parts  of  the  fum  together,  and 
if  it  agrees  with  the  fum,  it  is  right. 

EXAMPLES. 

T.  A  fchool-mafter,  being  afked  how  many  fcholars  he 
had,  faid,  If  I  had  as  many  more  as  I  now  have,  three 
quarters  as  many,  half  as  m*.ny,  one  fourth  and  one  eighth 
as  many,  I  fhould  then  have  435  :  Of  what  number  did 
his  fchool  confiil  ? 


POSITION.  277 

Suppofe  he  had  80      As  290  :  435   ::  80 

As  many=8o  80  120 

£  as  many=6o  • 1 20 

\  as  many=4O  2910)348010(120  Ans.  90 

\  as  many=2O  29  60 

g-  asmany=to  30 

58  15 

290  58  

Proof  435 
o 

2.  A  perfon  lent  his  friend  a  fum  of  money  unknown, 
to  receive  intereft  for  the  fame  at  6  per  cent  per  annum, 
fimple  intereft,  and  at  the  end  of  12  year  ,  received  for 
princ.pal  and  intereft  D86o  :  What  was  the  fum  lent  ? 

Ans. 


3-  A,  B  and  C  joined  their  flocks,  and  gained  0353 
I2\c.  of  which  A  ;ook  up  a  certain  fum,  B  took  up  four 
times  fo  much  as  A,  and  C,  five  times  fo  much  as  B  : 
What  lhare  of  the  gain  had  each? 

f  D    14  i2^c.   A's  fhare. 

Ans.  <        56   50        B's  fhare. 

I     282  50        C's  fhare. 

4.  A,  B,  C  and  D  fpent  355  at  a  reckoning,  and.  being 
a  little  dipped,  they  agreed  that  A  fhculd  pay  f,  B^,  C  -J-, 

What  did  each  pay  in  the  above  proportion  ? 

135.  4d. 

D 

C     6     8 
5     o 

5.  A  certain  fum  of  money  is  to  be  divided  between 
5  men,  in  fuch  a  manner  as  that  A  fhal!  have  ^.  B  J-,  C  r\y, 
D  3-5,  and  E  the  remainder,  which  is  ^"40  :     What  is  the 
fum  ? 

JSuppofe  £200.     Then  i+r+yo+sV^  ' 20* 

200 — 120=80.     As  80  :  40  ::   200  :   100  Ans, 

6.  A  and  B,  talking  of  their  ages,  B  faid  his  age  was 
once  and  an  half  the  age  of  A  ;  C  faid  his  was  twice  and 
one  tenth  the  age  of  both,  and  thn  the  fum  cf  their   ages 
was  93  :  What  was  the  age  of  each  ? 

Ans.  A's  1 2,  B's  i »,  and  C's  63  years. 


por 

[; 


*7&  POSITION. 

7.  A  and  B  having  found  a  purfe  of  money  difputed 
who  fhculd    have  it  :    A  faid  that  3-,  T\j-  and  ^1Q  of  it  a- 
mounted  to  £$$>  and,  if  B  could  tell  him  how  much  was 
in  it  he   (hould  have  the  whole,  otherwife  he   fiiould  have 
nothing  :  How  much  did  the  purfe  contain  ? 

Ans,  £100. 

8.  A  gentleman  divided  his   fortune  among  his    fons  ; 
to  A  he  gave  Dp  as  often  as  to  B  Dj,   and  to  C  but  D$ 
as  often  as    to  B  Dy,  yet  C's  portion  came  to  01059  : 
What  was  the  whole  eftate  ?  Ans.  07979  8oc. 

9.  Seven  eighths  of  a  certain  number  exceeds  four  fifths 
by  6  :  What  is  that  number  ?  Ans.  80. 

DOUBLE  POSITION. 

Double  Pofition  teaches  how  to  refolve  quefiions  by 
making  two  fuppofitions  of  falfe  numbers. 

Thofe  queftions,  in  which  the  refults  are  not  proportion* 
al  to  their  pofitions,  belong  to  this  rule  :  fuch  are  thole, 
in  which  the  number  fought  is  iricreafed  or  diminiflied  by 
fome  given  number,  which  is  no  known  part  of  the  num- 
ber required. 

Rutfc.* 

1.  Take   any  two  convenient  numbers,   and  proceed 
with  each  according  to  the  conditions  of  the  question. 

2.  Place  the  refult  or  errcurs  againft  their  pofitions  or 

Pos.       Err. 

30  —  ^  12 

fuppofed  numbers,  thus,         \f  and   if  the  errour 


fee  too  great,  mark  it  with  +  ;  and  if  too  frrall  with  —  . 

3.  Multiply  them  crofiwife  ;    that  is,  the  firil  pofition 
by  the  laft  errcur,  and  the  laft  pofition  by  the  firft  errour. 

4.  If  the  errours  be  alike,  that  is,  both  too  fmall  cr  both 
too  great,  divide  the  difference  of  the  products  by  the  dif- 
ference of  the  errours,  and  the  quotient  will  be  the  anfwer. 

*  The  rule  is  founded  on  this  fuppofition,  that  the  firft  errour  is 
to  the  fecond  as  the  difference  between  the  true  and  firft  fuppofed 
number  is  to  the  difference  between  the  true  and  leccnd  fuppofed 
number  :  When  that  is  not  the  cafe,  the  exa&  anfwer  to  the 
tion  cannot  befotindby  this  rule. 


POSITION.  279 

^.  If  the  errours  be  unlike,  that  is,  one  too  fmall,  and 
the  other  too  great,  divide  the  fum  of  the  produces  by  the 
fum  of  the  errours,  and  the  quotient  will  be  the  anfwer. 

Note.  When  the  errours  are  the  fame  in  quantity,  and 
unlike  in  quality,  half  the  fum  of  the  fuppofitions  is  the- 
numtier  fought. 

EXAMPLES. 

i.  A  lady  bought  damafk  for  a  gown,  at  8s.  per  yarcf, 
and  lining  for  it  at  33.  per  yard  ;  the  gown  and  lining 
contained  15  yards,  and  the  price  of  the/  whole  was  3!. 
i  os.  :  How  many  yards  were  there  of  each  ? 

Suppofe  6  yards  damafk,  value      483. 
Then  fhe  muS  have  9  yards  lining,  value     273. 

Sum  of  their  values=;5s. 
So  that  the  firft  errour  is  5  too  much,  or  4-     5 
Again,  fuppofe  fhe  had  4  yards  of  damafk,  value       325. 
Then  fhe  muft  have  1  1  yards  of  lining,  value       335. 

Sum  of  their  values=65s. 
So  that  the  fecond  errour  is  5  too  little,  or  —  5 


Then       ^V  5  yards  at  8*=£2     o  o 

4^  ^5  —    icyds  at  35.=   i    10  o 

20        30  Proof  3   10  o 

20 

Siim  of  errours=5+5=io)5o 

Ans.    5yds.  d.im  dk,  and   15-—  5= 
10  yards  lining.  Or  6+4-7-2=5  as  before. 

2.  A  labourer  was  hired  for  60  days,  upon  this  condition, 
that,  for  every  day  he  wrought,  he  ftiould  receive  75C.  ; 
and  for  every  day  he  was  idle,  fho-iM  torfeit  37jc.  ;  at 
the  expiration  of  the  time  he  received  Di8  :  How  many 
days  did  he  work,  and  how  many  was  he  idle  ? 


Suppofe  he  worked 

Ans.  He  was  employed  3  5  days,  and  was  idle  24. 


PERMUTATIONS  AND 

3.  A  gentleman  has  two  horfes  of  confiderable  value, 
and  a  carriage  worth  lool.  ;  now  if  the  fi-ft  horfe  be  bar- 
nelTed  in  it,  he  and  the  carriage  together  will  be  triple  the 
value  of  the  fecond  ;  but  if  the  fecond  be  put  in  they  will 
b?  7  times  the  value  of  the  iirft  :  Wnat  is  the  value  of 
each  horfe  ? 

f  32^7-8° — 

Suppofe    4        V^  Ans.  One  20!.  the  other  40!. 

(.44 -*••*•  1 60 — 

4  What  number  is  that,  which  being  increafed  by  its 
i>  its^,  and  5  mo/e,  will  be  doubled  ? 

Suppofe     \ 

(.16^-^1  +  Ans. 

5.  A  farmer  having  driven  his  cattle  to  market  receiv- 
ed for  them  all  0320,  being  paid  at  the  rate  of  1)24  per 
GX,  D 1 6  per  cow,  and  D6  per  calf;  there  was  as  many  oxen 
as  cows,  and  4  times  as  many  calves  as  cows  :  How 
many  were  there  of  each  fort  ? 

f   6  64* 

Suppofe    eows  -j        \f  Ans.  5  oxen,  5  cows, 

(.12  448-}-  and  20  calves. 


PERMUTATIONS  AND  COMBINATIONS, 

THE  Permutation  of  Quantities  is  the  (hewing  how 
many  different  ways  any  given  number  of  things  may  be 
changed. 

This  is  alfo  called  variation  alternation  or  changes  ;  and 
the  only  thing  to  be  regarded  here  is  the  order  they  (land 
in  ;  for  no  two  parcels  are  to  have  all  their  quantities 
placed  in  the  fame  fituation. 

The  Combination  of  Quantities  is  the  (hewing  how  of- 
ten a  lefs  number  of  things  can  be  taken  out  of  a  greater, 
and  combined  together,  without  confidering  their  places, 
or  the  order  they  ftand  in. 

This  is  fometimes  called  elettion  or  choice  ;  and  here 
every  parcel  muft  be  different  from  all  the  reft,  and  no  two 
are  te  have  precifely  the  fame  quantities,  or  things. 


COMBINATIONS.  281 

The  Composition  of  Quantities  is  the  taking  of  a  grv:n 
number  of  quantities,  ouc  of  as  many  equal  rows  of  dif- 
ferent quantities,  one  out  of  every  row,  and  combining 
them  together. 

Here  no  regard  is  had  to  their  places ;  and  it  differs 
from  Combination  only  as  that  admits  but  of  one  raw. 
of  things. 

PROBLEM    1. 

To  find  the  number  &f  permutations ,  or  changes,  that  can  be 
mads  of  any  given  number  of '  things ^  all  different  from  each 
othsr. 

RULE.* 

Multiply  all  the  terms  of  the  natural  feries  of  numbers, , 
from  i  up  to  the  given  number,  continually  together,  and 
the  laft  product  will  be  the  anfwer  required. 

EXAMPLES. 

1.  Chrift  church,  in  Bofton,  has  8  bells  :  How   many 
changes  may  he  rung  on  them  ? 

1X2X3X4X5X6X7X8=40320  Ans. 

2 .  Nine  gentlemen  met  at  an  inn,  and  were  fo  pleafed 
with  their  hoft,  and  with  each  other,  that  in  a  frolick,  they 
agreed  to  tarry  fo  long  as  they,   together  with   their  holt, 
could  fit  every  day  in  a  different  polition  at  dinner  :  Pray 
how  long,  had  they  kept  their  agreement,  would  their  frol- 
ick have  lafted  ?  Ans.  9^41^77  years. 

3.  How  many  changes,  or  variations,  will  the  alphabet 
admit  of  i.  Ans.  620448401733239439360000.. 

PROBLEM    II. 

Any  number  of  different  things  biing  ghen,  to  find  hoi*)  many 
changes  can  be  mj.de  out  of  them  by  taking  any  given  num~ 
ber  of  quantities  at  a  time. 

RULE. 
Take  a  feries  of  numbers,   beginning  at  the  number  of 

things  given,   and  decreaiing  by  i,  as  many  terms  as  the 

*  Any  two  things  <*  and  b  are  capable  of  two  variations  only  ;  as 
.«£,  ba ;  v/huie  uj.rKKr  ia  expreffed  by.  1  X'^- 

I:  there  be  t:iree  things  <i,  b  and  c  ,  then  any  two  of  them,  leavlm* 
out  the  tin:  i  1  X-  variations  ;  and  co.ifequently  w!  - 

third. is  'here  \vi!i  be.  1  X'-X-'   v^rhuions :  ;ud  ib  oo;  a^-lar 


282  PERMUTATIONS  AND 

number  of  quantities  to  be  taken  at  a  time  ;    the  proJudt 
of  all  the  terms  will  be  the  anfwer  required. 

EXAMPLES. 

i.  How  many  changes  may  be  rung  with  4  belUcut 
of  8  ? 

8 
7 

56 
6 

Or,  8X7X6X5  (=4  terms)  =1680  Ans. 


1680 

2.  How  many  words  can  be  made  with  6  letters  of  the 
alphabet,  admitting  a  number  of  confonants  may  make  a 
word  ? 

24X23X22X21x20X19  (6  terms)  =96909120,   Ans. 

PROBLEM    III. 

Any  number  of  things  being  given,  whereof  there  are  feveral 
things  of  one  forty  feveral  of  another^  &c.  to  find  koi»  many 
change*  may  be  mads  out  of  them  all. 

RULE.* 

i.  Take  the  feries  1X2X3x4,  &c.  up  to  the  number  of 
things  given,  and  find  the  produft  of  all  the  terms. 

*  Any  2  quantities,  a,  b,  both  different,  admit  of  2  changes;  but 
if  the  quantities  are  the  fame,  or  ab  becomes  aa  there  will  be  on- 

1X2 
ly  one  alteration,  which  may  be  exprefled  by  -  =1. 

1X2 

Any  3  quantities,  a,  b  c^  all  different  from  each  other,  admit  of  6 
variations  but  if  the  quantifies  are  all  alike,  or  abc  become  aaa,  then 
the  6  variations  will  be  reduced  to  1,  which  may  be  exprefled  by 
1X2X3 

--  -=1.       Again,  if  2  quantities  out  of  3  are  alike,  or  abc  be- 
,1X2X3 
come  aac  i   then  the  6  variations  will  be  reduced  to  thefe  3,  aac,  caat 

1X2X3 
tea,  which  may  be  exprefled  by  --  =3,  and  fo  of  any  greater 

1X2 
aumber. 


COMBINATIONS.  283 

2.  Take  the  feries    1X2X3X4,  Sec.  up  to  the  number  of 
the  given  things  of  the  firft  fort,  and  the  feries,  1X2X3X4, 
&c.  up  to  the  number  of  the  given  things  of  the  fecond 
fort,  &c. 

3.  Oivide  the  product  of  all  the  terms  of  the  firft  feries 
by  thcjoi.it  product  of  all  the  terms  of  the  remaining  ones, 
and  the  quotient  will  be  the  anfwer  required. 

EXAMPLES. 

1.  How  many  variations  may  be  made  of  the  letters  ia. 
the  word  2*mphnatbpaane«h  ? 

1X2X3x^x5x6x7x8X9X1^X1 1X12X1 3X14X15'  (=  number 
of  letters  in  the  word)  =1307674368000. 
1X2X3X4X5  (=number  of  as)  —120 
1X2  (=number  of  />s)  =     2 
i       (=number  of  ts\  =     i 
1X2X3  (=number  of  />s)  =     6 
1X2  (=number  of  »s)  =     2 
2X6X1X2X1 20=2880) 1 307674368000(454053600  Ans. 

2.  How  many  different  numbers  can  be  made  of  the 
following  figures,   1^3334444?  Ans.  12600. 


PROBLEM     IV. 

To  find  the  number  of  combination*  of  any  given  number  of 
things,  ail  different  from  one  another,  taken  any  given  num- 
ber at  a  time. 

RULE.* 

1.  Take  the  feries  i,  2,  3,  4   £s.  up  to  the   number  to 
be  taken  at  a  time,  and  find  the  product  of  all  the  terms.    • 

2.  Take  a  feries  of  as  rruny  terms,  decreafmg  by  i  from 
the  given  number,  out  of  which  the  election  is  to  be  made, 
and  find  the  product  of  all  the  terms. 

3.  Divide  the  laft  product  by  the  former,  and   the  quo- 
tient will  be  the  number  fought. 

*  In  any  given  number  of  quantities,  the  number  of  Combinations 
i  ncreafes  gradually  till  you  come  about  the  mean  numbers,  and  then 
gradually  deore  ties.  If  the  number  ot  qu  inline-)  be  ^v<n.  half  the  mi  11- 
ber  of  places  will  l"hew  the  great  eit  number  of  Combinations  that 
can  be  made  of  thofe  quantities  ;  but  if  odd,  then  thoie  two  numbers 
which  are  the  middle,  and  whofe  Cum  is  equal  to  the  given  number  ef 
quantities,  will  ihew  the  greateft  number  of  Combinations. 


284  PERMUTATIONS,  fcV. 

EXAMPLES. 

1.  How  many  combinations  may  be  made  of  7  letters 
out  of  12  ?. 

1X2X3X4X5X6x7  (=the  number  to  be  taken  at  a  time/ 
=5040. 

i 2X1 iXioX9X8X7X6(=  fame  number  from  12)^=3991680. 
5040)3991680(792  Ans. 

2.  How  many   combinations    can  be  made    of  6  let- 
ters out  of  the  24  letters  of  the  alphabet  * 

Ans.    134596. 

3.  A  farmer  bargained  with  a  gentleman  fcr  a  dozen 
fheep,  (at  2  dollars  per  head)  which  were  to  be  picked  out 
of  2  dozen  ;  but  being  long  choofing  them,  the  gentleman 
told  him  that  if  he  would  give  him  a  cent  for  every  dif- 
ierent  dozen  which  might  be  chofen  out  of  the  two  dozen, 
he  ihould  have  the  whole,  to  which  the   farmer    readily 
agreed  :  Pray  what  did  they  coft  him  ? 

Ans.  027041  560. 

PROBLEM     V. 

70  find  the  compojitiom  of  any  number •,  in  an  equal  number  of 
Jets,  the  things  being  all  different* 

RULE. 

Multiply  the  number  of  things  ia  every  fet  continually 
together,  and  the  product  will  be  the  anfwer  required. 

EXAMPLES. 

1.  Ruppofe  there  are  five  companies,  each  confining  of 
o  men  ;    it  is   required  to   find  how  many  >vays  5   men 
may  be  chcfen,  one  out  of  each  company  ? 

Multiply  9  into  itfelf  continually,  as  many  times  as 
there  are  companies. 

9X9X9X9X9—59049  different  ways,  Ans. 

2.  How  many   changes  are  there  in  throwing  4  dice  ? 
As  a  die  has  6  fiJes,  multiply  6  into  itfelf '  four  times 

continually.  6x6x6x6=1296  changes,    Ans. 

3  In  how  many  ways  may  a  m  tn,  a  woman  and  a 
child  be  choftn  out  of  thrve  companies,  confining  of  5 
men,  7  women  and  9  children  ?  Ans.  315. 


USE_OF  LOGARITHMS.  285- 


THE  USE  OF  LOGARITHMS. 

I.  Is  MULTIPLICATION. 

Given  two  numbers,  viz.  275  and  i2'6  to  find  their 
product. 

RULE.     To  the  logarithm  of  275,  viz.  2'43033 

Add  the  logarithm,  of  i  z  6,  viz.  i  •  0037 

And  their  fum  is  the  logarithm  7  _  — . 

of  their  producl,  viz.  34^5— 3'5397o 

2.     IN   DIVISION. 

Let  it  be  required  to  find  the  quotient,  which  arifes  by 
dividing  one  number  by  another  ;  fuppofe  1425  by  57. 

From  the  logarithm  of  the  dividend,  viz.  1425=3-15381 

Take  the  logarithm  of  the  divifor,  viz.  57  =  i  755*7 

And  the  remainder  is  the  logar-  1  • 

ithm  of  the  quotient,  viz.  25=1  39794. 

3.  IN   THE  RULE  OF  THREE. 
Three  numbers  given  to  find  a  fourth.  In  direfl  proportion. 

RULE.  From  the  tables  take  the  logarithms  of  each 
»f  the  propofed  numbers,  then  add  the  logarithms  of  the 
fecund  and  third  together,  and  from  the  fum  take  the 
logarithm  of  the  firft  and  the  remainder  will  be  the  logar 
rithm  of  the  fourth  number. 

Let  the  three  propofed  numbers  be  18,  24  and  33, 
and  the  operation  will  (land  thus  : 

1-38021  =  the  logarithm  of  24,  the  2d  term, 

i  5 185 1  =  the  logarithm  of  33,  the  3d  term. 

2-89872  =  the  logarithm  of  their  producl. 
—i '255 27  =  the  logarithm  of  the  lit  term  18. 

1-64345  =  the  logarithm  of  the  4th  term  required* 
which,  by  the  Table,  anfwers  to  the  natural  number  44^ 
the  4th  proportional  to  the  three  propofed  numbers, 


236  USE  OF  LOGARITHMS. 

4.  IN  INVOLUTION,  OR  RAISING  POWERS. 

To  find  any  p?<w?r  of  any  propofed  number,  or  to  twuofvt  any 

number  to  any  propofid  power  ^  by  logarithms. 
RULE.     Multiply  the  logarithm  of  the  given  root  by 
the  power,  viz.  by  2  for  the  fquare,  by  3  for  the  cubs,  &c. 
and  the  produdlis  the  logarithm  of  the  power  fought* 
Required  to  find  the  cube  of  12  ? 
107918*=  ihe  logarithm  of  1 2. 

X  3  =  the  third  powsr,  or  cube. 

3-23754^728  the  cube  of  12. 

5.   IN  EVOLUTION,  OR  EXTRACTING   ROOT$. 
To   extraft  any  rovt   of  any  propofsd  number. 

RULE.  Divide  the  logarithm  of  the  propefed  number 
by  the  index  of  the  required  root,  viz.  by  2  for  the  fqnare, 
by  3  for  the  cube,  &c.  and  the  quotient  will  be  the  loga- 
rithm of  the  root  required. 

Required  to  fin^  the  cube  root  of  172$  ? 

3'23754=:the  logarithm  of  1728,  *nd  3*23754-7-5*= 
1-07918  is  the  logarithm  of  the  cube  root  of  1728^ 
viz.  12. 

6.  IN  COMPOUND  INTEREST. 

To  fnd '  ths  amount  of  any  fan  for  any  time  and  at  any  rate,  at 
compound  intereji. 

RULE.  Multiply  the  logarithm  of  the  ratio  (i.  e.  the 
amount  of  f\  or'Di  for  I  year)  by  the  number  of  years> 
and  to  the  product  add  the  logarithm  of  the  principal  j 
the  fum  will  be  the  logarithm  of  the  amount, 
k  What  will  45!.  amount  to  forborne  12  years,  at  6 -per 
cent,  per  annum,  compound  intereft  ? 

Log.  of  1-06,  the  ratio  is  '02533  , 

Multiply  by  the  time  iz 

•30396 
Log.  of  45  the  principal   i '65321 

The  fum  is  1*95717      which   is    the 
logarithm  of  90  7  =£90  14^.  Ans,. 


RULES,  fcV.  287 

7.     IN   DISCOUNT  AT   COMPOUND    INTEREST. 

To  find  the  prefcnt  worth  of  any  fum    of  money,  due  any  time 
hence  >  at  any  rate >  at  compound  inter  eft. 

RULE.  From  the  logarithm  of  the  fum  to  be  difcount- 
ed,  fubtract  the  logarithm  of  the  rate  multiplied  by  the 
time  ;  and  the  remainder  is  the  logarithm  of  the  prefeut 
worth. 

What  prefent  money  will  pay  a  debt  of  90!.  145.  due 
1 1  years  hence,  difcounting  at  the  rate  of  6  per  cent,  per 
annum  ? 

From  the  logarithm  of  90!.   14=1*95717 

Sub.  producl  f.-f  ihe   Log.    uf  the"!  _  g 

ratio  X  by  the  tiir.e,  J 

The  remainder     1-65321 
is  the  logarithm  of  45!.  Ans. 


RULES 

For  reducing  Fedeffel  Money  to  the  federal  former  cur- 
rencies of  the  United  States  and  to  Englifh  Money,  and 
the  contrary. 

s.  d. 

f  6  o  old  currency  of  New  F,ngland  and  Virginia, 
j  8  o  New  York  and  N.  Carolina, 

l )  _  '  »  5  J  New    Jcrfcy,      Pennfylvania, 

I      Delaware  and  Maryland. 

j  4  8  South  Carolina  and  Georgia. 

1.4  6  Sterling  or  Engliih  Money. 

NOTE.  Mo  ft  of  the  operations  under  the  following 
rules  are  by  decimals ;  fo  that,  if  pounds  with  lower  de- 
nomipations  be  given,  the  lower  denominations  muft  be 
reduced  to  decimals  of  a  pound  ;  and,  if  dollars,  £c.  be 
reduced  Lo  pounds  and  decimals,  the  decimals  may  be  re- 
duced to  {killings,  &c«  in.  the  anfwer. 


288  RULES  FOR  REDUCING 

I.      To  reduce  Federal  Money  and  the   old  currency  of  Ne-w* 
England  and  Virginia  to  each  other. 

Multiply  dollars  by  '3,  or  divide  by  3  \  for  pounds. 
Multiply  pounds  by  j|,  or  divide  by  -3  for  dollars. 

EXAMPLES. 
!,  Reduce  D  150  to  pounds.         2.  Reduce  ^45  to  dollars. 

—  Jl 

£45-0  Ans.  135 

15 

Or,  34  =V°  ;  and,  - 

=i5oXT3o:i=T5TJ0*=  V  »5o  Ans. 

Ans.  as  before.  Or,  '3)45* 


Ans.  as  before. 

3.     Reduce    Di7     8ic.         4.  Reduce  £$   6s. 
to  pounds,  &c.  to  dollars,  &c. 


17  8*25 

4 

2-          qr 

*3 

12 

10-5       d. 

£5'34375 

_ 

20 

20 

6-875  s- 

s,  6-875 

£5'34375 

12 

d.  10-5 

16  03  125 

4 

1-78:25 

qr.  2fo  D 1 7-8125     Ans. 
Ans.  £5   6s.    i old. 

Or,  3-i=V°;    and   17-8125  Or,  '3)5*34375 

+- 17°=  1 7'8 .  Z5xr3*.  orx<3  ; 

hence  the  operation  and  re-  D  17*8 125  Ans.  as  be- 

fult  as  before,  fore. 

5.  Reduce  0271   I7C.  5m.  to  pounds,  &c. 

Ans.  £Si   7 

6.  Reduce  ^"8 1   73,  o^d.  to  dollars,  &c. 

Ans.  1)271   17*:,  5m.— • 


FEDERAL  MONEY, 


289 


7.  Reduce  Di3$  620.  5m.  to  pounds,  &c. 

Ans.  £41   us.  9d. 
$.  Reduce  £41   us.  pd.  to  dollars,  &c. 

Ans.  Di3&  6zc.  501. 


II.      71?   reduce  Federal   Money  to  the  old  currency  of 

Tork    and   North-Caroling  and  the  contrary. 
Multiply  dollars  by  "4,  or  divide  by  2-5,  for  pounds. 
Multiply  pounds  by  2'£,  or  divide  by   '4  for  dollars. 

EXAMPLES. 
i.  Reduce   59  dollars  to         2    Reduce  £23   125.  to 


pounds,  Sec. 


59 
'4 


20 
S.    12* 

Ans. 


I2S. 


Or,  2'5)59'o{23   12  Ans. 
50        as  before. 

90 
75 

15 
20 


dollars,  &c. 


— 
1180 
47* 


Ans. 


Or,  -4)23-6 


fore. 


Ans.  as  be- 


300 
25 

50 

5° 

3.  Reduce    D2i2    430.  4.  Reduce  £84  195. 

to  pounds,  &c.  to  dollars,  &c. 

375X-4=/,'84'975=  £84  1.95.    6d  =/* 

£84  193    6d.  /ins.  £84-975X2  5  =  D2i2'437£ 

Or,  2 1 2-4375 -1-2-5=  Ans. 

£84-975=^84  I9s.j5d.  Or,  £84-975~-4= 

Ans.  as  before.  0212-4375  Ans.  as  before; 
AA 


290  RULES  FOR  REDUCING 

5.  Reduce  074  3^0.  to  pounds,  &c. 

Ans   ,£29    145.  8d. 

6.  Reduce  £29    145;.   Sd,  to  dollars,  &c. 

Ans.  B74  33^c. 

III.      To  reduce  Federal  money  and  tie  old  currency  of  New. 
Jerfey*   Ptnnfyhania^   &c.  to  each  other. 

MuVJp'y  dollars  by  -375,  or  divide  by  .«£,  for  pounds. 
Multiply  pounds  by  2-f,  or  divide  by  -375  for  dollars. 

EXAMPLES. 

i.   Reduce  059  75c.    to          2    Reduce  £22   8s.  i|d. 
pounds,  &c.  to  dollars,  &c. 

^59*75  L22  %*'  i^'—L22'Ar®&2$ 

*375  2§ 

2^875  44-81250 

41825  .  14-9375 

17925  ~ 

—  Ans.  1)59-75 

£22   Ss     ijd.  Ans.    '  Or, 

Or,  2§s=J  ;    and  r>59'75       •375>-Hc62>(D59175 

179-25  Ans.  as  before. 


£22-40625  as  before. 

3.  Reduce  D79  870  5m.  to  pounds  &c. 

Ans.  £30  195.  o|d. 
4    Reduce  ^"30   195.  c|d.  to  dollars,  &c. 

Ars    ^79  870.  5rn. 

5.  Reduce  Di3i   8oc.  to  pounds,  &c. 

Ans.  ^49   8s,   6d. 

6.  Reduce  £49  8s.  6d.  to  collars,  &c. 

Ans.  1)11   8oc. 


IV.      To  reduce  Federal  Money  and  the  old  currency  of  South- 
Carolina  and  Georgia  to  each  other. 

Multiply  dollars  by  "2^,  cr  divide  by  4^,  for  pounds, 
Multiply  pounds  by   4^-,  or  divide  by  -2^,  for  dollars. 


FEDERAL  MONEY,  bV.  291 

EXAMPLES, 

f  T.   Reduce   375     dollars         2.   Reduce  £87    ics.  to 
to  pounds,  &c.    '  dollar?,      ~ 

375V 
' 


125  "'5^ 

£*rs  -  - 

20  ^375'     Ans. 

*7 

s.   io-  Or,  '2y-~  ;   and  £87  5 

Or,  &-  V°s  and  3 


2625  ~  —  ^37'5x—  ^  -- 

-  =87-5  as  before.  3  *7  -7 

30  =Dj75*  An£.  u^  before. 

3.  Reduce  D67  87:.  5rn.  to  pounds,  &j 

.-^ns    ^'15    163    gd. 

4.  Reduce  ^15    16:.  gd.  to  dollars,  &ci 

Ans.   1)67   S;?.   5-1. 

5.  Reduce  Dsoo  to  pounds,  &c.       Ans.  ^"23   C>   oJ. 

6.  Reduce  ^23  6s.  Sd.  lo  dolLirt,  Jcc.         Ar.s.  I; 

V.      Ti  rsducs  Federal  Money  to    EngllJI)    JLny,    .:::J   tbt 

•  con*  i. 

Multiply  dollars  by  '225,  or  divide  by  4^  for  pounds. 
Multiply  pounds  by  4^,  or  divide  by  -225,  far  dollars. 

EXAMPLES. 
i.  Reduce  D444  4fjc.  to  pounds. 


• 

222220 

888^8 


1.00=  J  cf  '225 
£100*00000  An:. 


-92  RULES,  e*V. 

Or,  4£=4»  ;  and  444-444=:l££°  .  then,  ±<LLt  .4.  *•  ~ 


-^r~  *  *V  *  3|f§°  =£100  Ans.  as  before, 
2.  Reduce  £100  to  dollars,  &c. 


400 


44*44$  =  I 


Or,  '225)ioo-(D444*44|  Ans,  as  before. 

3.  Reduce  Dzj  to  pounds,  &c. 

2^225=5-625  ;  and  ^5-625^5  125.  6d,  AHS. 

25 
Or,  — =  5-625,  as  before. 

4l 

4.  Reduce  £5   12$.  6d.  to  dollars. 

£5   i2s.  6d.=r£5-625  }   and  5-625X41=025  Ans, 
Or,   -225)5-625(025  Ans.  as    before. 

5.  Reduce  036  930.  7^m.  to  pounds,  &c. 

Ans.  £8  6s.  2d. 
i.  Reduce^S  6s.  2d,  a^qr.  to  dollars,  £c. 

Ans.  1)36  93c. 


USEFUL  TABLES. 


A  TABLE,  directing  how  to  buy  and  fell  by  th:  hurl- 
.  j,  in  FeJcr.J.    Money. 


D. 

D.      D. 

D.   ,    D. 

D. 

D.d. 

D.d. 

Dd. 

D.d.     Dd. 

D.d. 

(!.c. 

D-d.c. 

dc. 

D.dc.     d.c. 

D.dc. 

c.in. 

nd'cai. 

CiU 

D.dcm.     cm. 

D.dcm. 

1 

1  1  2 

34 

3808 

67 

7  5  ::'  i- 

2 

224 

35 

3920 

68 

7616 

3 

336 

4032 

69 

77^8 

4 

448 

37 

41  44 

7° 

7840 

5 

560 

4256 

?i 

o 

672 

39 

4368 

72 

8064 

7 

784 

40 

44#o 

73 

8:70 

8 

896 

41 

459^    74 

82bd 

9 

1008 

42 

4704    75 

8400 

10 

I  120 

43 

4816    76 

85^12 

1  1 

1232 

44 

4  v  -  3 

77 

8  )24 

12 

1344 

45 

50)0    78 

8  „-;  > 

1  3 

14^6 

46 

5*52    79 

8848 

14 

1568 

47 

5264    80 

8:/'ro 

15 

l68o 

48 

5  3  7  x>    8  1 

9072 

16 

1792 

49 

S4B8 

82 

91^4 

17 

1904 

5600 

8  j 

9296 

18 

20l6     51 

$712 

84 

'9 

2128     52 

5824 

85 

9520 

20 

2240 

53 

59-itf 

85 

1  '2 

21 

2352 

54 

;8 

87 

9744 

22 

2464 

55 

6:60 

9856 

23 

25/6 

627  i 

89 

24 

2688 

57 

6384 

9  ) 

ipt&a 

25 

2800 

5? 

6496 

9  r 

10192 

26 

2<}12 

59 

6608 

02 

1- 

27 

3024 

Oo 

6720 

TO.j 

28 

3  1  3  '* 

01 

6  642 

9  [ 

10528 

29 

32  }8 

62 

^44 

95 

»  0640 

30 

3363 

63 

765/5 

9  v 

107,2 

3  l 

3472 

64 

91 

I  C  'i  ;  ^ 

32 

35*4 

6.5 

72^0 

f 

1  C  .)  '  '> 

33 

3696 

61 

Pj 

1  K. 

AA    2 


294 


USEFUL  TABLES. 


A  TABLE  for  reducing  Troy  Weight  to  Avoirdupois. 


Troy.          Avoir. 

i  roy.    !               Avoiruupois. 

gr. 

drams. 

OZ. 

lb.       oz.    drams. 

1 

•04 

1 

1        1-55 

2 

•07 

2 

2       3-11 

3 

•11 

3 

3        4-66 

4 

•15 

4 

4        6-22 

5 

•18 

5 

5       7-77 

6 

•22 

6 

6        £-32 

7 

•26 

7 

7      1089 

8 

•29 

8 

8      1  2-44 

9 

•33 

9 

9      14- 

10 

•36 

10 

10      15-56 

11 

•4 

11 

12        1-09 

'          12 

•44 

lb. 

13 

•47 

1 

0     13        2-65 

14 

•51 

2 

1      10        5-3 

15 

•55 

3 

278- 

16 

•58 

4 

3       4     10-6 

17 

•62 

5 

4        1      13-25 

1         18 

•66 

6 

4     14      15-9 

19 

•69 

7 

5      12        2-56 

20 

•73 

8 

6        9       5-21 

21 

•77 

9 

7       <3        7*86 

22 

•S 

10 

8        3      10-52 

23 

•84 

20 

16        7        5-03 

pwt. 

30 

24     10     15-54 

1 

0-88 

40 

32      14     10-05 

2 

1-75 

50 

41        2        4-57 

3 

2-63 

60 

49       5      15-08 

4 

3-51 

70 

57        9        9-6 

5 

4.39 

80 

65      13        4-11 

6 

5-27 

90 

74        0      13-62 

7 

6-14 

100 

82       4        9-15 

'            S 

7-02 

200 

164        9       2-£8 

9             7-9 

300 

246      13     11-42 

!     10 

$.78 

400 

329       2       4-57 

'     11 

9-65 

500 

41  1        6      13-71 

12           10-53 

600 

493      1  1        C-85 

;       is       n-41 

700 

576       0       0- 

[          14      i      12-29 

800 

658        4        9-14 

I          15     i      13-16 

900 

740        9        2-28 

16      ;      M-04 

1000 

822      33      11-42 

17           14-92 

2000 

1645      11        6-84 

;>      is       15.79 

3000      :      25  f  8        9        2-26 

i            19             1  -  f  7                                                        fi       1  f>-'7° 

USEFUL  TABLES, 


TABLE  for  reducing  Avoirdupois  Weight  to  Troy 

Weight. 


Av. 

7™j'. 

Avoir. 

Troy. 

dr. 

ib.  oz.  pwt    gr. 

ib 

Ib.     tz.  /TWA  £r. 

Tt 

13-6; 

i 

I        2     II     1  6 

a 

4 

20-5] 

2 

2538 

1 

1    3-54 

3 

3     7   J5     o 

2 

2       6-681 

4 

4   10     6   16 

i% 

3 

3     ro'o2 

5 

6     o   18     8 

4 

5     »3'36 

6 

7     3    10     o 

5 

6     16  7 

7 

8     6     i    16 

6 

7     20*04 

8 

9     8   13     8 

7 

8    2338 

9 

ion     5     o 

8 

9      2'12 

10 

12     i    16  16 

9 

10      6  06 

20 

24     3   '3     8 

10 

ii      9-4 

3° 

36     5   10     o 

1  1 

12       I  2  '74 

40 

48     7     6  16 

1  2 

13     15-08 

>o 

60     9     3     8 

13 

14   19-42 

60 

7211     o     o 

14 

15   22-76 

70 

85     o   16   16 

'5 

17          21 

80 

97     2   13     8 

tfZ. 

90 

109     4  10     o 

i 

18      5-5 

I  00 

121     6     6  16 

2 

i    1  6    1  1 

2OO 

243     o  13     b 

3 

2   14    16-5 

300 

364     7     o     o 

4 

312       22 

4CO 

486     i     6   16 

5 

4   I[       3'5 

5CO 

607     7    13     8 

6 

599 

6co 

729       2       O       0 

/ 

6     7     14-5 

700 

850     8     6   16 

8 

7     5    20 

8co 

972     2   13     b 

9 

84       1-5 

900 

1093     9     o     o 

lO 

9     2       7 

1000 

i  2  •  5     3     616 

i 

10     o    12-5 

2003 

2430     6   13     8 

2 

IOl8     1  8 

3645    *o     o     o 

3 

II      I  6      2_-;*5 

4OOO 

4861      i     6    i  6 

i       o   15      5 

5OJO 

6076     4   13     8 

5 

I           T     1}      1O'? 

6000 

7291     8     o     o 

USEFUL  TABLES. 


By  a  law  of  the  Congrefs  of  the  United  States,  foreign 
,  gold  and  fi'ver  coins  are  made  a  legal  tender  for  the 
payment  of  all  debts  and  demands,  at  tht  fever  ai  und 
refpeclive  rates  following,  viz  The  gold  coins  ci 
Great  Britain  and  Portugal,  of  their  prdent  ft  a:' 
at  the  rate  cf  ico  cents  ior  every  27  gnilns  of  the  ic- 
tual  weight  thereof — Thole  of  France  and  .Sjj.i. 
grains  of  the  afiu  -I-  weight  thereof — Spaniih  ni.iud 
dollars,  weighing  i7p.vt.  70,rs-  equal  to  i  o  ce:i:s,  and 
in  proportion  for  the  p.n-ts  of  a  dolia; — Crowns  of 
Fiv.V'C-r,  \\-tigIiing  iBpwt.  l?gfs.  equal  to  i  ;  o  ccntr, 
and  in  r>iof  >t\io  i  f  >r  liie  j^artb  of  a  crown.  it  i:-  aiio 
enacted,  That  every  cent  ihali  contain  208  grains  of 
copper,  and  every  half  cent  104  grains  ;  agreeab'.y  to 
which  the  folio  wins:  Tables  are  calculated. 


A   TABLE  of  the  Weights  of  feveral  pieces  of  iinglifh, 
Portu;;uefe,  and    French  Go;d  Coins,   according  to  an 


acl  of  Congrefs,  pan^ci 

Nov. 

1792. 

|Pwt. 

Gr. 

Drs 

Cents 

Mills. 

Double  Johannes      18 

16 

o 

o 

Single  d^to                  9 

I 
i 

8 

o 

0 

Knglifh   Guinea          5 

C  i 

4 

66J 

.0 

Hair  ditto                    2 

'5  ! 

2 

33JF 

o 

French   Guinea           5 

6  j 

4 

59 

8 

H  df  ditto                   2 

15 

^ 

*9 

9 

4    P.    o  c»                   1  6 

i  2. 

14 

45 

2 

2   Piitoles                     S 

6 

7 

22 

6 

i    Phlole                       4 

s 

3 

6l 

3 

Moidore                        6 

2  2 

6 

'4 

8 

USEFUL  TABLES. 


A  TABLE  of  the  value  of  Englifh  and  Portuguefe 
Gold,  in  Dollars,  Cents,  and  Mills*  throughout  the 
United  States. 


Gr. 

Cu.  ra. 

Pwt. 

Dels.  Cts. 

I 

3   7 

I 

o   89 

a 

7   4 

2 

I     7"7-| 

3 

ii   i 

3 

2   66^ 

4 

14   8 

4 

3   55* 

5 

I^t    ° 

5 

4   44 

6 

22    2 

6 

5   33-J 

7 

25   9 

7 

6    22 

8 

29   6 

8 

7   II 

9 

33i  ° 

9 

8    o 

10 

37   ° 

10 

8   89 

ii 

40   7 

I  C 

9   77i 

12 

44   4 

12 

10   66^ 

13 

48   i 

13 

n   55^ 

14 

51   8^ 

H 

12   44! 

'5 

55   5? 

15 

13   33-J 

16 

59t  ° 

16 

14    22 

17 

63   o 

17 

15  II 

18 

66*  o 

18 

16    o 

19 

70   4 

'9 

16   89 

20 

74   o 

IOZ 

*7   77i 

21 

111   0 

22 

8i£  o 

23 

85    2 

} 

NOTE.     89  cents  is  the  value  of    i    pennyweight  of 
Englifli  and  Portuguefe  gold, 


USEFUL  TABLES. 


A  TABLE  of  the  value  of  French  anl  Spinifii  Gold, 
in  Dollars,  Cents,  and  Mills,  throughout  the  United 
States. 


Gr, 

Ccs      m. 

Pvvt. 

Dols.  Cts.  m 

i 

3       6 

i 

o     87       6 

2 

7       3 

2                   1       75          2 

3 

I  I          O 

3 

2     62^     o 

4 

14       6 

4 

3     5'^       3 

$ 

IS          2 

5 

4     38       o 

6 

21       9 

6 

5     25       5 

7 

25       5 

7 

6     13       i 

8 

29          2 

7 

707 

9 

32       8 

8 

7     88       3 

1Q 

36!     o 

9 

8     76       o 

1  I 

40       i 

10 

9     ^3i     ° 

!   2 

43       8 

n 

10     51        J 

•3 

47       4         i 

12 

ii      38       7 

5i        * 

13 

12     26       3 

15 

541     o 

J4 

13      13       9 

16 

58       4 

14 

14                     I  Tf              O 

17 

62       o 

15 

14     89       o 

18 

65       7 

16 

15     76       6 

!9 

69       3 

17 

16     64.       2 

20 

7.?       ° 

10Z 

17     51        8 

2  1 

76       6 

22 

80       3 

23 

S3       9 

NOTE.    87  cents  6  mills  is  the  value  of  i  penny \vsight 
French  and  Spanifh  Gold. 


FINfS. 


TABLE  OF  CONTENTS. 


Page 
NUMERATION  ,7 

Simple  Addition  10 

Subtraction  iz 

Multiplication  13 

•— — —  Dividcn  17 

Tables  of  coin,  weight,  and  meafure  az 

Compound  Addirion 

> —  Subtraction 

Problems  refultir,  •  ^mparifon  of  the  preceding  rules 

Application  of  the  preceding  rules 
Reduction   ;',  lending 

Alccnding 

Dcicending  and  Afcendiog 

Vulgar  Fractions 

Reduction  of  Vulgar  Fractions 

Addition  of         do. 

Subtraction  of      do. 

Multiplication  and  Divifion  cf  da. 

Decimal  Fractions 

Additi  >n  of-  Decimals 

Subtraction  of  Decimals 

Multiplication  of  Decimals 

Divifion  of  Decimals 

Reduction  of  Decimals 

Decimal  Tables  of  coin,  weight,  and  racafurt 

Compound  Multiplication 

Divifion 

Duodecimal? 

vSingle  Rule  of  Three  Dir  -ct: 

Rule  of  Three  Direct  in  Vulgar  Fractious. 

do.  in  Decimals 

Rule  of  Three  Inverfe 
Double  Rule  of  Three 
Conjoined  Proportion 
Arbitration  of  Exchangee 
Single  Fellow(hip, 
Double  Fellowfliip 
Fd!'j\v(hip  by  Decimals 


300  CONTENTS.  ' 

Practice 

Tare  and  Tret  ,5X 

Involution  ,6 

Table  of  Powers  l65 

Evolution  J6. 

The  Square  Root  5*0 

The  Cube  Root  ,1* 

The  Biquadrate  Root  !3£ 

Genera!  Rule  for  extracting  Roots  181 

Proportion  in  General  jg- 

Arithmetical  Propoition  jg4 

•-'—  Progreffion  j8< 

Geometrical  Proportion  I9I 

Progreffion  1Qa 

Simple  fntereft  ^QI 

Commiflion  /jc4 
Brokerage  3O4 

Buying  and  felling  Stocks  aoj 

Simple  Litereli  by  Decimals  aog 

in  Federal  Money  all 

Difcount  2ao 

Batter  aai 

Lofs  and  Gain  aa4 

Equation  of  Payments  -2^x 

Policies  of  Infurance  ajo 

Compound  intereft  a^? 

by  Decimals  a4a 

Difcount  by  Compound  Intereft  S4g 

Annuities,  Penfions,  &c.  a4Q 

Tables  a6; 

Alligation  a^9 

Pofition  a^5 

Permutations  and  Combinations  age 

Ufe  of  Logararithms  ag5 
R-ji.  s  for  reducing  federal  Money  to  feyeral  currendes  2,87 
KFfefal  TabJ»s 


*« 


I 


